L L is a high-level, open-source, general-purpose and system programming language which emphasizes readability, simplicity, extensibility, conciseness and performance. The L compiler features native code generation through LLVM, and is fully documented in a literate programming style. The language and compiler are usable, but are under heavy development as new features are being implemented.
L1-Norm Kernel PCA We present the first model and algorithm for L1-norm kernel PCA. While L2-norm kernel PCA has been widely studied, there has been no work on L1-norm kernel PCA. For this non-convex and non-smooth problem, we offer geometric understandings through reformulations and present an efficient algorithm where the kernel trick is applicable. To attest the efficiency of the algorithm, we provide a convergence analysis including linear rate of convergence. Moreover, we prove that the output of our algorithm is a local optimal solution to the L1-norm kernel PCA problem. We also numerically show its robustness when extracting principal components in the presence of influential outliers, as well as its runtime comparability to L2-norm kernel PCA. Lastly, we introduce its application to outlier detection and show that the L1-norm kernel PCA based model outperforms especially for high dimensional data.
Label Embedding Network We propose a method, called Label Embedding Network, which can learn label representation (label embedding) during the training process of deep networks. With the proposed method, the label embedding is adaptively and automatically learned through back propagation. The original one-hot represented loss function is converted into a new loss function with soft distributions, such that the originally unrelated labels have continuous interactions with each other during the training process. As a result, the trained model can achieve substantially higher accuracy and with faster convergence speed. Experimental results based on competitive tasks demonstrate the effectiveness of the proposed method, and the learned label embedding is reasonable and interpretable. The proposed method achieves comparable or even better results than the state-of-the-art systems. The source code is available at \url{https://…/LabelEmb}.
Labeled Latent Dirichlet Allocation
Labeled Latent Dirichlet Allocation (LLDA) is an extension of the standard unsupervised Latent Dirichlet Allocation (LDA) algorithm, to address multi-label learning tasks. Previous work has shown it to perform in par with other state-of-the-art multi-label methods. Nonetheless, with increasing label sets sizes LLDA encounters scalability issues. In this work, we introduce Subset LLDA, a simple variant of the standard LLDA algorithm, that not only can effectively scale up to problems with hundreds of thousands of labels but also improves over the LLDA state-of-the-art. We conduct extensive experiments on eight data sets, with label sets sizes ranging from hundreds to hundreds of thousands, comparing our proposed algorithm with the previously proposed LLDA algorithms (Prior–LDA, Dep–LDA), as well as the state of the art in extreme multi-label classification. The results show a steady advantage of our method over the other LLDA algorithms and competitive results compared to the extreme multi-label classification algorithms.
LAD Regression
Ladder The organizer of a machine learning competition faces the problem of maintaining an accurate leaderboard that faithfully represents the quality of the best submission of each competing team. What makes this estimation problem particularly challenging is its sequential and adaptive nature. As participants are allowed to repeatedly evaluate their submissions on the leaderboard, they may begin to overfit to the holdout data that supports the leaderboard. Few theoretical results give actionable advice on how to design a reliable leaderboard. Existing approaches therefore often resort to poorly understood heuristics such as limiting the bit precision of answers and the rate of re-submission. In this work, we introduce a notion of leaderboard accuracy tailored to the format of a competition. We introduce a natural algorithm called the Ladder and demonstrate that it simultaneously supports strong theoretical guarantees in a fully adaptive model of estimation, withstands practical adversarial attacks, and achieves high utility on real submission files from an actual competition hosted by Kaggle.
Lagrange Multiplier In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints.
Lagrange Policy Gradient Most algorithms for reinforcement learning work by estimating action-value functions. Here we present a method that uses Lagrange multipliers, the costate equation, and multilayer neural networks to compute policy gradients. We show that this method can find solutions to time-optimal control problems, driving nonlinear mechanical systems quickly to a target configuration. On these tasks its performance is comparable to that of deep deterministic policy gradient, a recent action-value method.
Lambda Architecture Lambda Architecture proposes a simpler, elegant paradigm that is designed to tame complexity while being able to store and effectively process large amounts of data. The Lambda Architecture was originally presented by Nathan Marz, who is well known in the big data community for his work on the Storm project.
Lambda Architecture
Lambda Architecture
LambdaMART At a high level, LambdaMART is an algorithm that uses gradient boosting to directly optimize Learning to Rank specific cost functions such as NDCG.
Lambert W Function In mathematics, the Lambert W function, also called the omega function or product logarithm, is a set of functions, namely the branches of the inverse relation of the function z = f(W) = We^W where e^W is the exponential function and W is any complex number. In other words, the defining equation for W(z) is: z = W(z)e^{W(z)} for any complex number z.
Lanczos Latent Factor Recommender
The purpose if this master’s thesis is to study and develop a new algorithmic framework for Collaborative Filtering to produce recommendations in the top-N recommendation problem. Thus, we propose Lanczos Latent Factor Recommender (LLFR); a novel ‘big data friendly’ collaborative filtering algorithm for top-N recommendation. Using a computationally efficient Lanczos-based procedure, LLFR builds a low dimensional item similarity model, that can be readily exploited to produce personalized ranking vectors over the item space. A number of experiments on real datasets indicate that LLFR outperforms other state-of-the-art top-N recommendation methods from a computational as well as a qualitative perspective. Our experimental results also show that its relative performance gains, compared to competing methods, increase as the data get sparser, as in the Cold Start Problem. More specifically, this is true both when the sparsity is generalized – as in the New Community Problem, a very common problem faced by real recommender systems in their beginning stages, when there is not sufficient number of ratings for the collaborative filtering algorithms to uncover similarities between items or users – and in the very interesting case where the sparsity is localized in a small fraction of the dataset – as in the New Users Problem, where new users are introduced to the system, they have not rated many items and thus, the CF algorithm can not make reliable personalized recommendations yet.
Lanczos Method The Lanczos algorithm is a direct algorithm devised by Cornelius Lanczos that is an adaptation of power methods to find the most useful eigenvalues and eigenvectors of an n t h {\displaystyle n^{th}} n^{{th}} order linear system with a limited number of operations, m {\displaystyle m} m, where m {\displaystyle m} m is much smaller than n {\displaystyle n} n. Although computationally efficient in principle, the method as initially formulated was not useful, due to its numerical instability. In 1970, Ojalvo and Newman showed how to make the method numerically stable and applied it to the solution of very large engineering structures subjected to dynamic loading. This was achieved using a method for purifying the vectors to any degree of accuracy, which when not performed, produced a series of vectors that were highly contaminated by those associated with the lowest natural frequencies. In their original work, these authors also suggested how to select a starting vector (i.e. use a random number generator to select each element of the starting vector) and suggested an empirically determined method for determining m {\displaystyle m} m, the reduced number of vectors (i.e. it should be selected to be approximately 1 ½ times the number of accurate eigenvalues desired). Soon thereafter their work was followed by Paige who also provided an error analysis. In 1988, Ojalvo produced a more detailed history of this algorithm and an efficient eigenvalue error test. Currently, the method is widely used in a variety of technical fields and has seen a number of variations.
Landmark Retracing Network
Since convolutional neural network (CNN) lacks an inherent mechanism to handle large scale variations, we always need to compute feature maps multiple times for multi-scale object detection, which has the bottleneck of computational cost in practice. To address this, we devise a recurrent scale approximation (RSA) to compute feature map once only, and only through this map can we approximate the rest maps on other levels. At the core of RSA is the recursive rolling out mechanism: given an initial map on a particular scale, it generates the prediction on a smaller scale that is half the size of input. To further increase efficiency and accuracy, we (a): design a scale-forecast network to globally predict potential scales in the image since there is no need to compute maps on all levels of the pyramid. (b): propose a landmark retracing network (LRN) to retrace back locations of the regressed landmarks and generate a confidence score for each landmark; LRN can effectively alleviate false positives due to the accumulated error in RSA. The whole system could be trained end-to-end in a unified CNN framework. Experiments demonstrate that our proposed algorithm is superior against state-of-the-arts on face detection benchmarks and achieves comparable results for generic proposal generation. The source code of RSA is available at
Langevin Monte Carlo
Language Model A statistical language model assigns a probability to a sequence of m words by means of a probability distribution. Language modeling is used in many natural language processing applications such as speech recognition, machine translation, part-of-speech tagging, parsing and information retrieval.
Large Vocabulary Continuous Speech Recognition System
The search problem in LVCSR can be simply stated: find the most probable sequence of words given a sequence of acoustic observations, an acoustic model and a language model. This is a demanding problem since word boundary information is not available in continuous speech and each word in the dictionary may be hypothesized to start at each frame of acoustic data. The problem is further complicated by the vocabulary size (typically 65,000 words) and the structure imposed on the search space by the language model. Direct evaluation of all the possible word sequences is impossible (given the large vocabulary) and an efficient search algorithm will consider only a very small subset of all possible utterance models. Typically, the effective size of the search space is reduced through pruning of unlikely hypotheses and/or the elimination of repeated computations.
Large-Scale Information Network Embedding
This paper studies the problem of embedding very large information networks into low-dimensional vector spaces, which is useful in many tasks such as visualization, node classification, and link prediction. Most existing graph embedding methods do not scale for real world information networks which usually contain millions of nodes. In this paper, we propose a novel network embedding method called the ‘LINE,’ which is suitable for arbitrary types of information networks: undirected, directed, and/or weighted. The method optimizes a carefully designed objective function that preserves both the local and global network structures. An edge-sampling algorithm is proposed that addresses the limitation of the classical stochastic gradient descent and improves both the effectiveness and the efficiency of the inference. Empirical experiments prove the effectiveness of the LINE on a variety of real-world information networks, including language networks, social networks, and citation networks. The algorithm is very efficient, which is able to learn the embedding of a network with millions of vertices and billions of edges in a few hours on a typical single machine. The source code of the LINE is available online.
Largest Gaps In this paper, the algorithm $Largest$ $Gaps$ is introduced, for simultaneously clustering both rows and columns of a matrix to form homogeneous blocks. The definition of clustering is model-based: clusters and data are generated under the Latent Block Model. In comparison with algorithms designed for this model, the major advantage of the $Largest$ $Gaps$ algorithm is to cluster using only some marginals of the matrix, the size of which is much smaller than the whole matrix. The procedure is linear with respect to the number of entries and thus much faster than the classical algorithms. It simultaneously selects the number of classes as well, and the estimation of the parameters is then made very easily once the classification is obtained. Moreover, the paper proves the procedure to be consistent under the LBM, and it illustrates the statistical performance with some numerical experiments.
Lasagne In this work we propose Lasagne, a methodology to learn locality and structure aware graph node embeddings in an unsupervised way. In particular, we show that the performance of existing random-walk based approaches depends strongly on the structural properties of the graph, e.g., the size of the graph, whether the graph has a flat or upward-sloping Network Community Profile (NCP), whether the graph is expander-like, whether the classes of interest are more k-core-like or more peripheral, etc. For larger graphs with flat NCPs that are strongly expander-like, existing methods lead to random walks that expand rapidly, touching many dissimilar nodes, thereby leading to lower-quality vector representations that are less useful for downstream tasks. Rather than relying on global random walks or neighbors within fixed hop distances, Lasagne exploits strongly local Approximate Personalized PageRank stationary distributions to more precisely engineer local information into node embeddings. This leads, in particular, to more meaningful and more useful vector representations of nodes in poorly-structured graphs. We show that Lasagne leads to significant improvement in downstream multi-label classification for larger graphs with flat NCPs, that it is comparable for smaller graphs with upward-sloping NCPs, and that is comparable to existing methods for link prediction tasks.
Lasso Penalized Sparse Asymmetric Least Squares
Lasso Regression
Last Observation Projection
LASVM is an approximate SVM solver that uses online approximation. It reaches accuracies similar to that of a real SVM after performing a single sequential pass through the training examples. Further benefits can be achieved using selective sampling techniques to choose which example should be considered next. As show in the graph, LASVM requires considerably less memory than a regular SVM solver. This becomes a considerable speed advantage for large training sets. In fact LASVM has been used to train a 10 class SVM classifier with 8 million examples on a single processor.
Latent Association Mining in Binary Data
We consider the problem of identifying groups of mutually associated variables in moderate or high dimensional data. In many cases, ordinary Pearson correlation provides useful information concerning the linear relationship between variables. However, for binary data, ordinary correlation may lose power and may lack interpretability. In this paper, we develop and investigate a new method called Latent Association Mining in Binary Data (LAMB). The LAMB method is built on the assumption that the binary observations represent a random thresholding of a latent continuous variable that may have a complex correlation structure. We consider a new measure of association, latent correlation, that is designed to assess association in the underlying continuous variable, without bias due to the mediating effects of the thresholding procedure. The full LAMB procedure makes use of iterative hypothesis testing to identify groups of latently correlated variables. LAMB is shown to improve power over existing methods in simulated settings, to be computationally efficient for large datasets, and to uncover new meaningful results from common real data types.
Latent Attention Network Deep neural networks are able to solve tasks across a variety of domains and modalities of data. Despite many empirical successes, we lack the ability to clearly understand and interpret the learned internal mechanisms that contribute to such effective behaviors or, more critically, failure modes. In this work, we present a general method for visualizing an arbitrary neural network’s inner mechanisms and their power and limitations. Our dataset-centric method produces visualizations of how a trained network attends to components of its inputs. The computed ‘attention masks’ support improved interpretability by highlighting which input attributes are critical in determining output. We demonstrate the effectiveness of our framework on a variety of deep neural network architectures in domains from computer vision, natural language processing, and reinforcement learning. The primary contribution of our approach is an interpretable visualization of attention that provides unique insights into the network’s underlying decision-making process irrespective of the data modality.
Latent Class Analysis
Latent class analysis (LCA) identifies unobservable subgroups within a population.
Latent Class Model
In statistics, a latent class model (LCM) relates a set of observed (usually discrete) multivariate variables to a set of latent variables. It is a type of latent variable model. It is called a latent class model because the latent variable is discrete. A class is characterized by a pattern of conditional probabilities that indicate the chance that variables take on certain values. Latent Class Analysis (LCA) is a subset of structural equation modeling, used to find groups or subtypes of cases in multivariate categorical data. These subtypes are called “latent classes”. Confronted with a situation as follows, a researcher might choose to use LCA to understand the data: Imagine that symptoms a-d have been measured in a range of patients with diseases X Y and Z, and that disease X is associated with the presence of symptoms a, b, and c, disease Y with symptoms b, c, d, and disease Z with symptoms a, c and d. The LCA will attempt to detect the presence of latent classes (the disease entities), creating patterns of association in the symptoms. As in factor analysis, the LCA can also be used to classify case according to their maximum likelihood class membership. Because the criterion for solving the LCA is to achieve latent classes within which there is no longer any association of one symptom with another (because the class is the disease which causes their association, and the set of diseases a patient has (or class a case is a member of) causes the symptom association, the symptoms will be “conditionally independent”, i.e., conditional on class membership, they are no longer related.
Latent Constrained Correlation Filter
Correlation filters are special classifiers designed for shift-invariant object recognition, which are robust to pattern distortions. The recent literature shows that combining a set of sub-filters trained based on a single or a small group of images obtains the best performance. The idea is equivalent to estimating variable distribution based on the data sampling (bagging), which can be interpreted as finding solutions (variable distribution approximation) directly from sampled data space. However, this methodology fails to account for the variations existed in the data. In this paper, we introduce an intermediate step — solution sampling — after the data sampling step to form a subspace, in which an optimal solution can be estimated. More specifically, we propose a new method, named latent constrained correlation filters (LCCF), by mapping the correlation filters to a given latent subspace, and develop a new learning framework in the latent subspace that embeds distribution-related constraints into the original problem. To solve the optimization problem, we introduce a subspace based alternating direction method of multipliers (SADMM), which is proven to converge at the saddle point. Our approach is successfully applied to three different tasks, including eye localization, car detection and object tracking. Extensive experiments demonstrate that LCCF outperforms the state-of-the-art methods. The source code will be publicly available. https://…/.
Latent Constraints Deep generative neural networks have proven effective at both conditional and unconditional modeling of complex data distributions. Conditional generation enables interactive control, but creating new controls often requires expensive retraining. In this paper, we develop a method to condition generation without retraining the model. By post-hoc learning latent constraints, value functions that identify regions in latent space that generate outputs with desired attributes, we can conditionally sample from these regions with gradient-based optimization or amortized actor functions. Combining attribute constraints with a universal ‘realism’ constraint, which enforces similarity to the data distribution, we generate realistic conditional images from an unconditional variational autoencoder. Further, using gradient-based optimization, we demonstrate identity-preserving transformations that make the minimal adjustment in latent space to modify the attributes of an image. Finally, with discrete sequences of musical notes, we demonstrate zero-shot conditional generation, learning latent constraints in the absence of labeled data or a differentiable reward function. Code with dedicated cloud instance has been made publicly available ( ).
Latent Dirichlet Allocation
In natural language processing, latent Dirichlet allocation (LDA) is a generative model that allows sets of observations to be explained by unobserved groups that explain why some parts of the data are similar. For example, if observations are words collected into documents, it posits that each document is a mixture of a small number of topics and that each word’s creation is attributable to one of the document’s topics. LDA is an example of a topic model and was first presented as a graphical model for topic discovery by David Blei, Andrew Ng, and Michael Jordan in 2003.
Latent Dirichlet allocation Gibbs Newton
Hyper-parameters play a major role in the learning and inference process of latent Dirichlet allocation (LDA). In order to begin the LDA latent variables learning process, these hyper-parameters values need to be pre-determined. We propose an extension for LDA that we call ‘Latent Dirichlet allocation Gibbs Newton’ (LDA-GN), which places non-informative priors over these hyper-parameters and uses Gibbs sampling to learn appropriate values for them. At the heart of LDA-GN is our proposed ‘Gibbs-Newton’ algorithm, which is a new technique for learning the parameters of multivariate Polya distributions. We report Gibbs-Newton performance results compared with two prominent existing approaches to the latter task: Minka’s fixed-point iteration method and the Moments method. We then evaluate LDA-GN in two ways: (i) by comparing it with standard LDA in terms of the ability of the resulting topic models to generalize to unseen documents; (ii) by comparing it with standard LDA in its performance on a binary classification task.
Latent Factor Interpretation
Many machine learning systems utilize latent factors as internal representations for making predictions. However, since these latent factors are largely uninterpreted, predictions made using them are opaque. Collaborative filtering via matrix factorization is a prime example of such an algorithm that uses uninterpreted latent features, and yet has seen widespread adoption for many recommendation tasks. We present Latent Factor Interpretation (LFI), a method for interpreting models by leveraging interpretations of latent factors in terms of human-understandable features. The interpretation of latent factors can then replace the uninterpreted latent factors, resulting in a new model that expresses predictions in terms of interpretable features. This new model can then be interpreted using recently developed model explanation techniques. In this paper, we develop LFI for collaborative filtering based recommender systems, which are particularly challenging from an interpretation perspective. We illustrate the use of LFI interpretations on the MovieLens dataset demonstrating that latent factors can be predicted with enough accuracy for accurately replicating the predictions of the true model. Further, we demonstrate the accuracy of interpretations by applying the methodology to a collaborative recommender system using DB tropes and IMDB data and synthetic user preferences.
Latent Feature Relational Model
We present a discriminative nonparametric latent feature relational model (LFRM) for link prediction to automatically infer the dimensionality of latent features. Under the generic RegBayes (regularized Bayesian inference) framework, we handily incorporate the prediction loss with probabilistic inference of a Bayesian model; set distinct regularization parameters for different types of links to handle the imbalance issue in real networks; and unify the analysis of both the smooth logistic log-loss and the piecewise linear hinge loss. For the nonconjugate posterior inference, we present a simple Gibbs sampler via data augmentation, without making restricting assumptions as done in variational methods. We further develop an approximate sampler using stochastic gradient Langevin dynamics to handle large networks with hundreds of thousands of entities and millions of links, orders of magnitude larger than what existing LFRM models can process. Extensive studies on various real networks show promising performance.
Latent Gaussian Process Regression We introduce Latent Gaussian Process Regression which is a latent variable extension allowing modelling of non-stationary processes using stationary GP priors. The approach is built on extending the input space of a regression problem with a latent variable that is used to modulate the covariance function over the input space. We show how our approach can be used to model non-stationary processes but also how multi-modal or non-functional processes can be described where the input signal cannot fully disambiguate the output. We exemplify the approach on a set of synthetic data and provide results on real data from geostatistics.
Latent Semantic Analysis
Latent semantic analysis (LSA) is a technique in natural language processing, in particular in vectorial semantics, of analyzing relationships between a set of documents and the terms they contain by producing a set of concepts related to the documents and terms. LSA assumes that words that are close in meaning will occur in similar pieces of text. A matrix containing word counts per paragraph (rows represent unique words and columns represent each paragraph) is constructed from a large piece of text and a mathematical technique called singular value decomposition (SVD) is used to reduce the number of columns while preserving the similarity structure among rows. Words are then compared by taking the cosine of the angle between the two vectors formed by any two rows. Values close to 1 represent very similar words while values close to 0 represent very dissimilar words.
Latent Sequence Decompositions
We present the Latent Sequence Decompositions (LSD) framework. LSD decomposes sequences with variable lengthed output units as a function of both the input sequence and the output sequence. We present a training algorithm which samples valid extensions and an approximate decoding algorithm. We experiment with the Wall Street Journal speech recognition task. Our LSD model achieves 12.9% WER compared to a character baseline of 14.8% WER. When combined with a convolutional network on the encoder, we achieve 9.2% WER.
Latent Structure Analysis
latent structure analysis (LSA). LSA is a broad category that subsumes several individual methods, including latent class analysis (LCA) and latent trait analysis (LTA). The purpose of LSA is to infer, from observed variables (manifest variables), the structure of other, more fundamental variables that are not directly observed (latent variables). Both manifest variables and latent variables can be binary, nominal, ordered-categorical, or interval/continuous – leading to a large different combinations and different methods. For example, classical latent class analysis considers binary, nominal, or ordered-categorical manifest variables and nominal latent variables, and latent trait analysis considers binary or ordered-categorical variables and continuous latent variables.
Latent Structure Learning
Latent Trait Analysis
Latent Trait Analysis (LTA), a form of latent structure analysis (Lazarsfeld & Henry, 1968), is used for the analysis of categorical data. The simplest way to understand it is that LTA is form of factor analysis for binary (dichotomous) or ordered-category data. In the area of educational testing and psychological measurement, latent trait analysis is termed Item Response Theory (IRT). There is so much overlap between LTA and IRT that these terms are basically interchangeable.


Latent Transition Analysis
Latent transition analysis (LTA) and latent class analysis (LCA) are closely related methods. LCA identifies unobservable (latent) subgroups within a population based on individuals’ responses to multiple observed variables. LTA is an extension of LCA that uses longitudinal data to identify movement between the subgroups over time.
Latent Tree Models Latent tree models are graphical models defined on trees, in which only a subset of variables is observed. They were first discussed by Judea Pearl as tree-decomposable distributions to generalise star-decomposable distributions such as the latent class model. Latent tree models, or their submodels, are widely used in: phylogenetic analysis, network tomography, computer vision, causal modeling, and data clustering. They also contain other well-known classes of models like hidden Markov models, Brownian motion tree model, the Ising model on a tree, and many popular models used in phylogenetics. This article offers a concise introduction to the theory of latent tree models. We emphasise the role of tree metrics in the structural description of this model class, in designing learning algorithms, and in understanding fundamental limits of what and when can be learned.
Latent Variable Mixture Model
Latent variable mixture modeling (LVMM) is a flexible analytic tool that allows researchers to investigate questions about patterns of data and to determine the extent to which identified patterns relate to important variables. For example,
• Do patterns of co-occurring developmental and medical diagnoses influence the severity of pediatric feeding problems (Berlin, Lobato, Pinkos, Cerezo, & LeLeiko, 2011)?
• Do differential longitudinal trajectories of glycemic control exist among youth with type 1 diabetes (Helgeson et al., 2010)
• Do differential trajectories of adherence among youth newly diagnosed with epilepsy exist (Modi, Rausch, & Glauser, 2011), and if so,
• Do psychosocial and demographic variables predict these patterns?
• Do patterns of perceived stressors among youth with type 1 diabetes differentially affect glycemic control (Berlin, Rabideau, & Hains, 2012)?
Latent Variable Model A latent variable model is a statistical model that relates a set of variables (so-called manifest variables) to a set of latent variables. It is assumed that the responses on the indicators or manifest variables are the result of an individual’s position on the latent variable(s), and that the manifest variables have nothing in common after controlling for the latent variable (local independence). Different types of the latent variable model can be grouped according to whether the manifest and latent variables are categorical or continuous.
Latin Hypercube Design MOLHD
Lavaan Project The lavaan package is developed to provide useRs, researchers and teachers a free open-source, but commercial-quality package for latent variable modeling. You can use lavaan to estimate a large variety of multivariate statistical models, including path analysis, confirmatory factor analysis, structural equation modeling and growth curve models. The official reference to the lavaan package is the following paper: Yves Rosseel (2012). lavaan: An R Package for Structural Equation Modeling. Journal of Statistical Software, 48(2), 1-36. URL http://…/i02
Law of Likelihood If hypothesis A implies that the probability that a random variable X takes the value x is pA(x), while hypothesis B implies that the probability is pB(x), then the observation X = x is evidence supporting A over B if and only if pA(x) > pB(x), and the likelihood ratio, pA(x)/ pB(x), measures the strength of that evidence.’
‘This says simply that if an event is more probable under hypothesis A than hypothesis B, then the occurrence of that event is evidence supporting A over B – the hypothesis that did the better job of predicting the event is better supported by its occurrence.’ Moreover, ‘the likelihood ratio, is the exact factor by which the probability ratio (ratio of priors in A and B) is changed.
Layer-Wise Relevance Propagation
Despite the tremendous achievements of deep convolutional neural networks~(CNNs) in most of computer vision tasks, understanding how they actually work remains a significant challenge. In this paper, we propose a novel two-step visualization method that aims to shed light on how deep CNNs recognize images and the objects therein. We start out with a layer-wise relevance propagation (LRP) step which estimates a pixel-wise relevance map over the input image. Following, we construct a context-aware saliency map from the LRP-generated map which predicts regions close to the foci of attention. We show that our algorithm clearly and concisely identifies the key pixels that contribute to the underlying neural network’s comprehension of images. Experimental results using the ILSVRC2012 validation dataset in conjunction with two well-established deep CNNs demonstrate that combining the LRP with the visual salience estimation can give great insight into how a CNNs model perceives and understands a presented scene, in relation to what it has learned in the prior training phase.
Lazy Bayesian Rules
The naive Bayesian classifier provides a simple and effective approach to classifier learning, but its attribute independence assumption is often violated in the real world. A number of approaches have sought to alleviate this problem. A Bayesian tree learning algorithm builds a decision tree, and generates a local naive Bayesian classifier at each leaf. The tests leading to a leaf can alleviate attribute inter-dependencies for the local naive Bayesian classifier. However, Bayesian tree learning still suffers from the small disjunct problem of tree learning. While inferred Bayesian trees demonstrate low average prediction error rates, there is reason to believe that error rates will be higher for those leaves with few training examples. This paper proposes the application of lazy learning techniques to Bayesian tree induction and presents the resulting lazy Bayesian rule learning algorithm, called Lbr. This algorithm can be justified by a variant of Bayes theorem which supports a weaker conditional attribute independence assumption than is required by naive Bayes. For each test example, it builds a most appropriate rule with a local naive Bayesian classifier as its consequent. It is demonstrated that the computational requirements of Lbr are reasonable in a wide cross-section of natural domains. Experiments with these domains show that, on average, this new algorithm obtains lower error rates significantly more often than the reverse in comparison to a naive Bayesian classifier, C4.5, a Bayesian tree learning algorithm, a constructive Bayesian classifier that eliminates attributes and constructs new attributes using Cartesian products of existing nominal attributes, and a lazy decision tree learning algorithm. It also outperforms, although the result is not statistically significant, a selective naive Bayesian classifier.
Lazy Learning In artificial intelligence, lazy learning is a learning method in which generalization beyond the training data is delayed until a query is made to the system, as opposed to in eager learning, where the system tries to generalize the training data before receiving queries. The main advantage gained in employing a lazy learning method, such as Case based reasoning, is that the target function will be approximated locally, such as in the k-nearest neighbor algorithm. Because the target function is approximated locally for each query to the system, lazy learning systems can simultaneously solve multiple problems and deal successfully with changes in the problem domain. The disadvantages with lazy learning include the large space requirement to store the entire training dataset. Particularly noisy training data increases the case base unnecessarily, because no abstraction is made during the training phase. Another disadvantage is that lazy learning methods are usually slower to evaluate, though this is coupled with a faster training phase. Lazy classifiers are most useful for large datasets with few attributes.
Lazy Stochastic Principal Component Analysis
(Lazy SPCA)
Stochastic principal component analysis (SPCA) has become a popular dimensionality reduction strategy for large, high-dimensional datasets. We derive a simplified algorithm, called Lazy SPCA, which has reduced computational complexity and is better suited for large-scale distributed computation. We prove that SPCA and Lazy SPCA find the same approximations to the principal subspace, and that the pairwise distances between samples in the lower-dimensional space is invariant to whether SPCA is executed lazily or not. Empirical studies find downstream predictive performance to be identical for both methods, and superior to random projections, across a range of predictive models (linear regression, logistic lasso, and random forests). In our largest experiment with 4.6 million samples, Lazy SPCA reduced 43.7 hours of computation to 9.9 hours. Overall, Lazy SPCA relies exclusively on matrix multiplications, besides an operation on a small square matrix whose size depends only on the target dimensionality.
lda2vec Standard natural language processing (NLP) is a messy and difficult affair. It requires teaching a computer about English-specific word ambiguities as well as the hierarchical, sparse nature of words in sentences. At Stitch Fix, word vectors help computers learn from the raw text in customer notes. Our systems need to identify a medical professional when she writes that she ‘used to wear scrubs to work’, and distill ‘taking a trip’ into a Fix for vacation clothing. Applied appropriately, word vectors are dramatically more meaningful and more flexible than current techniques and let computers peer into text in a fundamentally new way. I’ll try to convince you that word vectors give us a simple and flexible platform for understanding text while speaking about word2vec, LDA, and introduce our hybrid algorithm lda2vec.
Leader Clustering Algorithm The leader clustering algorithm provides a means for clustering a set of data points. Unlike many other clustering algorithms it does not require the user to specify the number of clusters, but instead requires the approximate radius of a cluster as its primary tuning parameter. The package provides a fast implementation of this algorithm in n-dimensions using Lp-distances (with special cases for p=1,2, and infinity) as well as for spatial data using the Haversine formula, which takes latitude/longitude pairs as inputs and clusters based on great circle distances.
Leaders and Subleaders Algorithm An efficient hierarchical clustering algorithm, suitable for large data sets is proposed for effective clustering and prototype selection for pattern classification. It is another simple and efficient technique which uses incremental clustering principles to generate a hierarchical structure for finding the subgroups/subclusters within each cluster. As an example, a two level clustering algorithm – Leaders-Subleaders, an extension of the leader algorithm is presented. Classification accuracy (CA) obtained using the representatives generated by the Leaders-Subleaders method is found to be better than that of using leaders as representatives. Even if more number of prototypes are generated, classification time is less as only a part of the hierarchical structure is searched.
leaflet Leaflet is a modern open-source JavaScript library for mobile-friendly interactive maps. It is developed by Vladimir Agafonkin with a team of dedicated contributors. Weighing just about 33 KB of JS, it has all the features most developers ever need for online maps. Leaflet is designed with simplicity, performance and usability in mind. It works efficiently across all major desktop and mobile platforms out of the box, taking advantage of HTML5 and CSS3 on modern browsers while still being accessible on older ones. It can be extended with a huge amount of plugins, has a beautiful, easy to use and well-documented API and a simple, readable source code that is a joy to contribute to.
Leaflet: Interactive web maps with R
Lean Analytics Lean Analytics is about measuring the right thing, in the right way, to produce the change the business needs the most at that point in time. With that in mind, here’s some background on metrics that matter.
Learning Active Learning
In this paper, we suggest a novel data-driven approach to active learning: Learning Active Learning (LAL). The key idea behind LAL is to train a regressor that predicts the expected error reduction for a potential sample in a particular learning state. By treating the query selection procedure as a regression problem we are not restricted to dealing with existing AL heuristics; instead, we learn strategies based on experience from previous active learning experiments. We show that LAL can be learnt from a simple artificial 2D dataset and yields strategies that work well on real data from a wide range of domains. Moreover, if some domain-specific samples are available to bootstrap active learning, the LAL strategy can be tailored for a particular problem.
Learning Analytics
Learning analytics is the measurement, collection, analysis and reporting of data about learners and their contexts, for purposes of understanding and optimising learning and the environments in which it occurs. A related field is educational data mining. For general audience introductions, see:
• The Educause Learning Initiative Briefing
• The Educause Review on Learning analytics
• And the UNESCO “Learning Analytics Policy Brief” (2012)
Learning Automata Based SVM
As an indispensable defensive measure of network security, the intrusion detection is a process of monitoring the events occurring in a computer system or network and analyzing them for signs of possible incidents. It is a classifier to judge the event is normal or malicious. The information used for intrusion detection contains some redundant features which would increase the difficulty of training the classifier for intrusion detection and increase the time of making predictions. To simplify the training process and improve the efficiency of the classifier, it is necessary to remove these dispensable features. in this paper, we propose a novel LA-SVM scheme to automatically remove redundant features focusing on intrusion detection. This is the first application of learning automata for solving dimension reduction problems. The simulation results indicate that the LA-SVM scheme achieves a higher accuracy and is more efficient in making predictions compared with traditional SVM.
Learning by Association In many real-world scenarios, labeled data for a specific machine learning task is costly to obtain. Semi-supervised training methods make use of abundantly available unlabeled data and a smaller number of labeled examples. We propose a new framework for semi-supervised training of deep neural networks inspired by learning in humans. ‘Associations’ are made from embeddings of labeled samples to those of unlabeled ones and back. The optimization schedule encourages correct association cycles that end up at the same class from which the association was started and penalizes wrong associations ending at a different class. The implementation is easy to use and can be added to any existing end-to-end training setup. We demonstrate the capabilities of learning by association on several data sets and show that it can improve performance on classification tasks tremendously by making use of additionally available unlabeled data. In particular, for cases with few labeled data, our training scheme outperforms the current state of the art on SVHN.
Learning Classifier System
Learning classifier systems, or LCS, are a paradigm of rule-based machine learning methods that combine a discovery component (e.g. typically a genetic algorithm) with a learning component (performing either supervised learning, reinforcement learning, or unsupervised learning).[2] Learning classifier systems seek to identify a set of context-dependent rules that collectively store and apply knowledge in a piecewise manner in order to make predictions (e.g. behavior modeling,[3] classification,[4][5] data mining,[5][6][7] regression,[8] function approximation,[9] or game strategy). This approach allows complex solution spaces to be broken up into smaller, simpler parts. The founding concepts behind learning classifier systems came from attempts to model complex adaptive systems, using rule-based agents to form an artificial cognitive system (i.e. artificial intelligence).
Learning Curve Plots relating performance to experience are widely used in machine learning. Performance is the error rate or accuracy of the learning system, while experience may be the number of training examples used for learning or the number of iterations used in optimizing the system model parameters. The machine learning curve is useful for many purposes including comparing different algorithms, choosing model parameters during design, adjusting optimization to improve convergence, and determining the amount of data used for training.
Learning M-Way Tree
LMW-tree is a generic template library written in C++ that implements several algorithms that use the m-way nearest neighbor tree structure to store their data. See the related PhD thesis for more details on m-way nn trees. The algorithms and data structures are generic to support different data representations such as dense real valued and bit vectors, and sparse vectors. Additionally, it can index any object type that can form a prototype representation of a set of objects. The algorithms are primarily focussed on computationally efficient clustering. Clustering is an unsupervised machine learning process that finds interesting patterns in data. It places similar items into clusters and dissimilar items into different clusters. The data structures and algorithms can also be used for nearest neighbor search, supervised learning and other machine learning applications. The package includes EM-tree, K-tree, k-means, TSVQ, repeated k-means, clustering, random projections, random indexing, hashing, bit signatures. See the related PhD thesis for more details these algorithms and representations. LMW-tree is licensed under the BSD license.
Learning Solving Procedure It is expected that progress toward true artificial intelligence will be achieved through the emergence of a system that integrates representation learning and complex reasoning (LeCun et al. 2015). In response to this prediction, research has been conducted on implementing the symbolic reasoning of a von Neumann computer in an artificial neural network (Graves et al. 2016; Graves et al. 2014; Reed et al. 2015). However, these studies have many limitations in realizing neural-symbolic integration (Jaeger. 2016). Here, we present a new learning paradigm: a learning solving procedure (LSP) that learns the procedure for solving complex problems. This is not accomplished merely by learning input-output data, but by learning algorithms through a solving procedure that obtains the output as a sequence of tasks for a given input problem. The LSP neural network system not only learns simple problems of addition and multiplication, but also the algorithms of complicated problems, such as complex arithmetic expression, sorting, and Hanoi Tower. To realize this, the LSP neural network structure consists of a deep neural network and long short-term memory, which are recursively combined. Through experimentation, we demonstrate the efficiency and scalability of LSP and its validity as a mechanism of complex reasoning.
Learning Vector Quantization
In computer science, learning vector quantization (LVQ), is a prototype-based supervised classification algorithm. LVQ is the supervised counterpart of vector quantization systems.
Learning with Counts
Learning with Opponent-Learning Awareness
Multi-agent settings are quickly gathering importance in machine learning. Beyond a plethora of recent work on deep multi-agent reinforcement learning, hierarchical reinforcement learning, generative adversarial networks and decentralized optimization can all be seen as instances of this setting. However, the presence of multiple learning agents in these settings renders the training problem non-stationary and often leads to unstable training or undesired final results. We present Learning with Opponent-Learning Awareness (LOLA), a method that reasons about the anticipated learning of the other agents. The LOLA learning rule includes an additional term that accounts for the impact of the agent’s policy on the anticipated parameter update of the other agents. We show that the LOLA update rule can be efficiently calculated using an extension of the likelihood ratio policy gradient update, making the method suitable for model-free reinforcement learning. This method thus scales to large parameter and input spaces and nonlinear function approximators. Preliminary results show that the encounter of two LOLA agents leads to the emergence of tit-for-tat and therefore cooperation in the infinitely iterated prisoners’ dilemma, while independent learning does not. In this domain, LOLA also receives higher payouts compared to a naive learner, and is robust against exploitation by higher order gradient-based methods. Applied to infinitely repeated matching pennies, only LOLA agents converge to the Nash equilibrium. We also apply LOLA to a grid world task with an embedded social dilemma using deep recurrent policies. Again, by considering the learning of the other agent, LOLA agents learn to cooperate out of selfish interests.
Least Absolute Deviations
Least absolute deviations (LAD), also known as Least Absolute Errors (LAE), Least Absolute Value (LAV), or Least Absolute Residual (LAR) or the L1 norm problem, is a statistical optimization technique similar to the popular least squares technique that attempts to find a function which closely approximates a set of data. In the simple case of a set of (x,y) data, the approximation function is a simple ‘trend line’ in two-dimensional Cartesian coordinates. The method minimizes the sum of absolute errors (SAE) (the sum of the absolute values of the vertical ‘residuals’ between points generated by the function and corresponding points in the data). The least absolute deviations estimate also arises as the maximum likelihood estimate if the errors have a Laplace distribution.
Least Absolute Shrinkage and Screening Operator
Slide 31: ‘Tibshirani (1996):
LASSO = Least Absolute Shrinkage and Selection Operator
new translation:
LASSO = Least Absolute Shrinkage and Screening Operator’
Least Absolute Shrinkage and Selection Operator
The Lasso is a shrinkage and selection method for linear regression. It minimizes the usual sum of squared errors, with a bound on the sum of the absolute values of the coefficients. It has connections to soft-thresholding of wavelet coefficients, forward stagewise regression, and boosting methods.
Least Square Projection
The problem of projecting multidimensional data into lower dimensions has been pursued by many researchers due to its potential application to data analysis of various kinds. This paper presents a novel multidimensional projection technique based on least square approximations. The approximations compute the coordinates of a set of projected points based on the coordinates of a reduced number of control points with defined geometry. We name the technique Least Square Projections (LSP).
Least Squares Deep Q-Network
Deep reinforcement learning (DRL) methods such as the Deep Q-Network (DQN) have achieved state-of-the-art results in a variety of challenging, high-dimensional domains. This success is mainly attributed to the power of deep neural networks to learn rich domain representations for approximating the value function or policy. Batch reinforcement learning methods with linear representations, on the other hand, are more stable and require less hyper parameter tuning. Yet, substantial feature engineering is necessary to achieve good results. In this work we propose a hybrid approach — the Least Squares Deep Q-Network (LS-DQN), which combines rich feature representations learned by a DRL algorithm with the stability of a linear least squares method. We do this by periodically re-training the last hidden layer of a DRL network with a batch least squares update. Key to our approach is a Bayesian regularization term for the least squares update, which prevents over-fitting to the more recent data. We tested LS-DQN on five Atari games and demonstrate significant improvement over vanilla DQN and Double-DQN. We also investigated the reasons for the superior performance of our method. Interestingly, we found that the performance improvement can be attributed to the large batch size used by the LS method when optimizing the last layer.
Least-Angle Regression
In statistics, least-angle regression (LARS) is a regression algorithm for high-dimensional data, developed by Bradley Efron, Trevor Hastie, Iain Johnstone and Robert Tibshirani. Suppose we expect a response variable to be determined by a linear combination of a subset of potential covariates. Then the LARS algorithm provides a means of producing an estimate of which variables to include, as well as their coefficients. Instead of giving a vector result, the LARS solution consists of a curve denoting the solution for each value of the L1 norm of the parameter vector. The algorithm is similar to forward stepwise regression, but instead of including variables at each step, the estimated parameters are increased in a direction equiangular to each one’s correlations with the residual.
Leave-One-Out Cross Validation
Leave-one-out cross-validation (LOOCV) is a particular case of leave-p-out cross-validation with p = 1.
Leave-p-Out Cross Validation
As the name suggests, leave-p-out cross-validation (LpO CV) involves using p observations as the validation set and the remaining observations as the training set. This is repeated on all ways to cut the original sample on a validation set of p’ observations and a training set. LpO cross-validation requires to learn and validate times (where n is the number of observation in the original sample). So as soon as n is quite big it becomes impossible to calculate.
Lemmatization Lemmatisation (or lemmatization) in linguistics is the process of grouping together the different inflected forms of a word so they can be analysed as a single item. In computational linguistics, lemmatisation is the algorithmic process of determining the lemma for a given word. Since the process may involve complex tasks such as understanding context and determining the part of speech of a word in a sentence (requiring, for example, knowledge of the grammar of a language) it can be a hard task to implement a lemmatiser for a new language. In many languages, words appear in several inflected forms. For example, in English, the verb ‘to walk’ may appear as ‘walk’, ‘walked’, ‘walks’, ‘walking’. The base form, ‘walk’, that one might look up in a dictionary, is called the lemma for the word. The combination of the base form with the part of speech is often called the lexeme of the word. Lemmatisation is closely related to stemming. The difference is that a stemmer operates on a single word without knowledge of the context, and therefore cannot discriminate between words which have different meanings depending on part of speech. However, stemmers are typically easier to implement and run faster, and the reduced accuracy may not matter for some applications.
Lempel-Ziv-Oberhumer (LZO) is a lossless data compression algorithm that is focused on decompression speed.
Lenstra Lenstra Lovasz
The Lenstra-Lenstra-Lovász (LLL) lattice basis reduction algorithm is a polynomial time lattice reduction algorithm invented by Arjen Lenstra, Hendrik Lenstra and László Lovász in 1982.
Levenberg-Marquardt Algorithm
In mathematics and computing, the Levenberg-Marquardt algorithm (LMA), also known as the damped least-squares (DLS) method, is used to solve non-linear least squares problems. These minimization problems arise especially in least squares curve fitting. The LMA interpolates between the Gauss-Newton algorithm (GNA) and the method of gradient descent. The LMA is more robust than the GNA, which means that in many cases it finds a solution even if it starts very far off the final minimum. For well-behaved functions and reasonable starting parameters, the LMA tends to be a bit slower than the GNA. LMA can also be viewed as Gauss-Newton using a trust region approach. The LMA is a very popular curve-fitting algorithm used in many software applications for solving generic curve-fitting problems. However, as for many fitting algorithms, the LMA finds only a local minimum, which is not necessarily the global minimum.
Levenshtein Distance In information theory and computer science, the Levenshtein distance is a string metric for measuring the difference between two sequences. Informally, the Levenshtein distance between two words is the minimum number of single-character edits (i.e. insertions, deletions or substitutions) required to change one word into the other. The phrase edit distance is often used to refer specifically to Levenshtein distance. It is named after Vladimir Levenshtein, who considered this distance in 1965. It is closely related to pairwise string alignments.
Lexical Dispersion Plot A Lexical Dispersion Plot shows the position of words in a given text. On the y axis there is the list of words to be looked at and on the x axis there is the position in the text. Therefore the highest value on the x axis is the lenght of the text.
Lexical Table
Lexis Surface Map
LexVec In this paper, we propose LexVec, a new method for generating distributed word representations that uses low-rank, weighted factorization of the Positive Point-wise Mutual Information matrix via stochastic gradient descent, employing a weighting scheme that assigns heavier penalties for errors on frequent co-occurrences while still accounting for negative co-occurrence. Evaluation on word similarity and analogy tasks shows that LexVec matches and often outperforms state-of-the-art methods on many of these tasks.
libDirectional In this paper, we present libDirectional, a MATLAB library for directional statistics and directional estimation. It supports a variety of commonly used distributions on the unit circle, such as the von Mises, wrapped normal, and wrapped Cauchy distributions. Furthermore, various distributions on higher-dimensional manifolds such as the unit hypersphere and the hypertorus are available. Based on these distributions, several recursive filtering algorithms in libDirectional allow estimation on these manifolds. The functionality is implemented in a clear, well-documented, and object-oriented structure that is both easy to use and easy to extend.
libFM Factorization machines (FM) are a generic approach that allows to mimic most factorization models by feature engineering. This way, factorization machines combine the generality of feature engineering with the superiority of factorization models in estimating interactions between categorical variables of large domain. libFM is a software implementation for factorization machines that features stochastic gradient descent (SGD) and alternating least squares (ALS) optimization as well as Bayesian inference using Markov Chain Monte Carlo (MCMC).
LibLinear LibLinear is a linear classifier for data with millions of instances and features. It supports
• L2-regularized classifiers L2-loss linear SVM,
• L1-loss linear SVM, and logistic regression (LR)
• L1-regularized classifiers (after version 1.4)
• L2-loss linear SVM and logistic regression (LR)
• L2-regularized support vector regression (after version 1.9)
• L2-loss linear SVR and L1-loss linear SVR.
Library for Online Learning
LIBOL is an open-source library for large-scale online learning, which consists of a large family of e cient and scalable state-of-the-art online learning algorithms for large-scale online classification tasks. We have offered easy-to-use command-line tools and examples for users and developers, and also have made comprehensive documents available for both beginners and advanced users. LIBOL is not only a machine learning toolbox, but also a comprehensive experimental platform for conducting online learning research.
Lifelong Learning
This paper proposes a novel lifelong learning (LL) approach to sentiment classification. LL mimics the human continuous learning process, i.e., retaining the knowledge learned from past tasks and use it to help future learning. In this paper, we first discuss LL in general and then LL for sentiment classification in particular. The proposed LL approach adopts a Bayesian optimization framework based on stochastic gradient descent. Our experimental results show that the proposed method outperforms baseline methods significantly, which demonstrates that lifelong learning is a promising research direction.
Lift In data mining and association rule learning, lift is a measure of the performance of a targeting model (association rule) at predicting or classifying cases as having an enhanced response (with respect to the population as a whole), measured against a random choice targeting model. A targeting model is doing a good job if the response within the target is much better than the average for the population as a whole. Lift is simply the ratio of these values: target response divided by average response. For example, suppose a population has an average response rate of 5%, but a certain model (or rule) has identified a segment with a response rate of 20%. Then that segment would have a lift of 4.0 (20%/5%). Typically, the modeller seeks to divide the population into quantiles, and rank the quantiles by lift. Organizations can then consider each quantile, and by weighing the predicted response rate (and associated financial benefit) against the cost, they can decide whether to market to that quantile or not. Lift is analogous to information retrieval’s average precision metric, if one treats the precision (fraction of the positives that are true positives) as the target response probability. The lift curve can also be considered a variation on the receiver operating characteristic (ROC) curve, and is also known in econometrics as the Lorenz or power curve. The difference between the lifts observed on two different subgroups is called the uplift. The subtraction of two lift curves forms the uplift curve, which is a metric used in uplift modelling. It is important to note that in general marketing practice the term Lift is also defined as the difference in response rate between the treatment and control groups, indicating the causal impact of a marketing program (versus not having it as in the control group). As a result, ‘no lift’ often means there is no statistically significant effect of the program. On top of this, uplift modelling is a predictive modeling technique to improve (up) lift over control.
Lift Chart The lift chart provides a visual summary of the usefulness of the information provided by one or more statistical models for predicting a binomial (categorical) outcome variable (dependent variable); for multinomial (multiple-category) outcome variables, lift charts can be computed for each category. Specifically, the chart summarizes the utility that we may expect by using the respective predictive models, as compared to using baseline information only. The lift chart is applicable to most statistical methods that compute predictions (predicted classifications) for binomial or multinomial responses.
Let us start with an example. A marketing agency is planning to send advertisements to selected households with the goal to boost sales of a product. The agency has a list of all households where each household is described by a set of attributes. Each advertisement sent costs a few pennies, but it is well paid off if the customer buys the product. Therefore an agency wants to minimize the number of advertisements sent, while at the same time maximize the number of sold products by reaching only the consumers that will actually buy the product. Therefore it develops a classifier that predicts the probability that a household is a potential customer. To fit this classifier and to express the dependency between the costs and the expected benefit the lift chart can be used. The number off all potential customers P is often unknown, therefore TPrate cannot be computed and the ROC curve cannot used, but the lift chart is useful in such settings. Also the TP is often hard to measure in practice; one might have just a few measurements from a sales analysis. Even in such cases lift chart can help the agency select the amount of most promising households to which an advertisement should be sent. Of course, lift charts are also useful for many other similar problems.
A lift chart, sometimes called a cumulative gains chart, or a banana chart, is a measure of model performance. It shows how responses, (i.e., to a direct mail solicitation, or a surgical treatment for instance) are changed by applying the model. This change ratio, which is hopefully, the increase in response rate, is called the ‘lift’. A lift chart indicates which subset of the dataset contains the greatest possible proportion of positive responses. The higher the lift curve is from the baseline, the better the performance of the model since the baseline represents the null model, which is no model at all. To explain a lift chart, suppose we had a two-class prediction where the outcomes were yes (a positive response) or no (a negative response). To create a lift chart, instances in the dataset are sorted in descending probability order according to the predicted probability of a positive response. When the data is plotted, we can see a graphical depiction of the various probabilities. While the example shown in Figure 10 plots the results of different datasets for a single model, a lift chart can also be used to plot the results of a single dataset for different models. Note that the best model is not the one with the highest lift when it is being built. It is the model that performs the best on unseen, future data.
Light Recurrent Neural Networks
Recurrent neural networks (RNNs) have achieved state-of-the-art performances in many natural language processing tasks, such as language modeling and machine translation. However, when the vocabulary is large, the RNN model will become very big (e.g., possibly beyond the memory capacity of a GPU device) and its training will become very inefficient. In this work, we propose a novel technique to tackle this challenge. The key idea is to use 2-Component (2C) shared embedding for word representations. We allocate every word in the vocabulary into a table, each row of which is associated with a vector, and each column associated with another vector. Depending on its position in the table, a word is jointly represented by two components: a row vector and a column vector. Since the words in the same row share the row vector and the words in the same column share the column vector, we only need $2 \sqrt{|V|}$ vectors to represent a vocabulary of $|V|$ unique words, which are far less than the $|V|$ vectors required by existing approaches. Based on the 2-Component shared embedding, we design a new RNN algorithm and evaluate it using the language modeling task on several benchmark datasets. The results show that our algorithm significantly reduces the model size and speeds up the training process, without sacrifice of accuracy (it achieves similar, if not better, perplexity as compared to state-of-the-art language models). Remarkably, on the One-Billion-Word benchmark Dataset, our algorithm achieves comparable perplexity to previous language models, whilst reducing the model size by a factor of 40-100, and speeding up the training process by a factor of 2. We name our proposed algorithm \emph{LightRNN} to reflect its very small model size and very high training speed.
LightFM Model
I present a hybrid matrix factorisation model representing users and items as linear combinations of their content features’ latent factors. The model outperforms both collaborative and content-based models in cold-start or sparse interaction data scenarios (using both user and item metadata), and performs at least as well as a pure collaborative matrix factorisation model where interaction data is abundant. Additionally, feature embeddings produced by the model encode semantic information in a way reminiscent of word embedding approaches, making them useful for a range of related tasks such as tag recommendations.
Likelihood Likelihood is a funny concept. It’s not a probability, but it is proportional to a probability. The likelihood of a hypothesis (H) given some data (D) is proportional to the probability of obtaining D given that H is true, multiplied by an arbitrary positive constant (K). In other words, L(H|D) = K · P(D|H). Since a likelihood isn’t actually a probability it doesn’t obey various rules of probability. For example, likelihood need not sum to 1. A critical difference between probability and likelihood is in the interpretation of what is fixed and what can vary. In the case of a conditional probability, P(D|H), the hypothesis is fixed and the data are free to vary. Likelihood, however, is the opposite. The likelihood of a hypothesis, L(H|D), conditions on the data as if they are fixed while allowing the hypotheses to vary. The distinction is subtle, so I’ll say it again. For conditional probability, the hypothesis is treated as a given and the data are free to vary. For likelihood, the data are a given and the hypotheses vary.


Likelihood Function In statistics, a likelihood function (often simply the likelihood) is a function of the parameters of a statistical model. The likelihood of a set of parameter values, theta, given outcomes x, is equal to the probability of those observed outcomes given those parameter values, that is L(theta|x) = P(x|theta).
Likelihood functions play a key role in statistical inference, especially methods of estimating a parameter from a set of statistics. In informal contexts, “likelihood” is often used as a synonym for “probability.” But in statistical usage, a distinction is made depending on the roles of the outcome or parameter. Probability is used when describing a function of the outcome given a fixed parameter value. For example, if a coin is flipped 10 times and it is a fair coin, what is the probability of it landing heads-up every time? Likelihood is used when describing a function of a parameter given an outcome. For example, if a coin is flipped 10 times and it has landed heads-up 10 times, what is the likelihood that the coin is fair?
Likelihood Ratio Similarity
Recommender system data presents unique challenges to the data mining, machine learning, and algorithms communities. The high missing data rate, in combination with the large scale and high dimensionality that is typical of recommender systems data, requires new tools and methods for efficient data analysis. Here, we address the challenge of evaluating similarity between two users in a recommender system, where for each user only a small set of ratings is available. We present a new similarity score, that we call LiRa, based on a statistical model of user similarity, for large-scale, discrete valued data with many missing values. We show that this score, based on a ratio of likelihoods, is more effective at identifying similar users than traditional similarity scores in user-based collaborative filtering, such as the Pearson correlation coefficient. We argue that our approach has significant potential to improve both accuracy and scalability in collaborative filtering.
Likelihood-Ratio Test
In statistics, a likelihood ratio test is a statistical test used to compare the fit of two models, one of which (the null model) is a special case of the other (the alternative model). The test is based on the likelihood ratio, which expresses how many times more likely the data are under one model than the other. This likelihood ratio, or equivalently its logarithm, can then be used to compute a p-value, or compared to a critical value to decide whether to reject the null model in favour of the alternative model. When the logarithm of the likelihood ratio is used, the statistic is known as a log-likelihood ratio statistic, and the probability distribution of this test statistic, assuming that the null model is true, can be approximated using Wilks’s theorem. In the case of distinguishing between two models, each of which has no unknown parameters, use of the likelihood ratio test can be justified by the Neyman-Pearson lemma, which demonstrates that such a test has the highest power among all competitors.
Likert Scale A Likert scale is a psychometric scale commonly involved in research that employs questionnaires. It is the most widely used approach to scaling responses in survey research, such that the term is often used interchangeably with rating scale, or more accurately the Likert-type scale, even though the two are not synonymous. The scale is named after its inventor, psychologist Rensis Likert. Likert distinguished between a scale proper, which emerges from collective responses to a set of items (usually eight or more), and the format in which responses are scored along a range. Technically speaking, a Likert scale refers only to the former.
Limited Memory Steepest Descent Method
The possibilities inherent in steepest descent methods have been considerably amplified by the introduction of the Barzilai-Borwein choice of step-size, and other related ideas. These methods have proved to be competitive with conjugate gradient methods for the minimization of large dimension unconstrained minimization problems. This paper suggests a method which is able to take advantage of the availability of a few additional ‘long’ vectors of storage to achieve a significant improvement in performance, both for quadratic and non-quadratic objective functions. It makes use of certain Ritz values related to the Lanczos process (Lanczos in J Res Nat Bur Stand 45:255-282, 1950). Some underlying theory is provided, and numerical evidence is set out showing that the new method provides a competitive and more simple alternative to the state of the art l-BFGS limited memory method.
Limited-memory BFGS
Limited-memory BFGS (L-BFGS or LM-BFGS) is an optimization algorithm in the family of quasi-Newton methods that approximates the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm using a limited amount of computer memory. It is a popular algorithm for parameter estimation in machine learning.
Lindy Effect The Lindy effect is a theory of the life expectancy of non-perishable things that posits for a certain class of nonperishables, like a technology or an idea, every additional day may imply a longer (remaining) life expectancy: the mortality rate decreases with time. This contrasts with living creatures and mechanical things, which instead follow a bathtub curve, where every additional day in its life translates into a shorter additional life expectancy (though longer overall life expectancy, due to surviving this far): after childhood, the mortality rate increases with time.
Line Map linemap
Linear Additive Markov Process
We introduce LAMP: the Linear Additive Markov Process. Transitions in LAMP may be influenced by states visited in the distant history of the process, but unlike higher-order Markov processes, LAMP retains an efficient parametrization. LAMP also allows the specific dependence on history to be learned efficiently from data. We characterize some theoretical properties of LAMP, including its steady-state and mixing time. We then give an algorithm based on alternating minimization to learn LAMP models from data. Finally, we perform a series of real-world experiments to show that LAMP is more powerful than first-order Markov processes, and even holds its own against deep sequential models (LSTMs) with a negligible increase in parameter complexity.
Linear Algebra Package
LAPACK (Linear Algebra Package) is a software library for numerical linear algebra. It provides routines for solving systems of linear equations and linear least squares, eigenvalue problems, and singular value decomposition. It also includes routines to implement the associated matrix factorizations such as LU, QR, Cholesky and Schur decomposition.
Linear Analog Self-Assessment
Linear Centralization Classifier
A classification algorithm, called the Linear Centralization Classifier (LCC), is introduced. The algorithm seeks to find a transformation that best maps instances from the feature space to a space where they concentrate towards the center of their own classes, while maximimizing the distance between class centers. We formulate the classifier as a quadratic program with quadratic constraints. We then simplify this formulation to a linear program that can be solved effectively using a linear programming solver (e.g., simplex-dual). We extend the formulation for LCC to enable the use of kernel functions for non-linear classification applications. We compare our method with two standard classification methods (support vector machine and linear discriminant analysis) and four state-of-the-art classification methods when they are applied to eight standard classification datasets. Our experimental results show that LCC is able to classify instances more accurately (based on the area under the receiver operating characteristic) in comparison to other tested methods on the chosen datasets. We also report the results for LCC with a particular kernel to solve for synthetic non-linear classification problems.
Linear Congruential Generator
A linear congruential generator (LCG) is an algorithm that yields a sequence of pseudo-randomized numbers calculated with a discontinuous piecewise linear equation. The method represents one of the oldest and best-known pseudorandom number generator algorithms. The theory behind them is relatively easy to understand, and they are easily implemented and fast, especially on computer hardware which can provide modulo arithmetic by storage-bit truncation.
Linear Dimension Reduction Methods:
1. Principal component analysis (PCA)
2. Canonical correlation analysis (CCA)
3. Linear discriminant analysis (LDA)
4. Non-negative matrix factorization (NMF)
5. Independent component analysis (ICA)
Linear Discriminant Analysis
Linear discriminant analysis (LDA) and the related Fisher’s linear discriminant are methods used in statistics, pattern recognition and machine learning to find a linear combination of features which characterizes or separates two or more classes of objects or events. The resulting combination may be used as a linear classifier, or, more commonly, for dimensionality reduction before later classification.
Fisher’s Linear Discriminant Analysis
Linear Discriminant Generative Adversarial Networks
We develop a novel method for training of GANs for unsupervised and class conditional generation of images, called Linear Discriminant GAN (LD-GAN). The discriminator of an LD-GAN is trained to maximize the linear separability between distributions of hidden representations of generated and targeted samples, while the generator is updated based on the decision hyper-planes computed by performing LDA over the hidden representations. LD-GAN provides a concrete metric of separation capacity for the discriminator, and we experimentally show that it is possible to stabilize the training of LD-GAN simply by calibrating the update frequencies between generators and discriminators in the unsupervised case, without employment of normalization methods and constraints on weights. In the class conditional generation tasks, the proposed method shows improved training stability together with better generalization performance compared to WGAN that employs an auxiliary classifier.
Linear Mixed Effects Model CLME,lmenssp
Linear Mixed Model
A statistical model containing both fixed effects and random effects, that is: mixed effects. LMM is a kind of regression analysis.
Linear Programming
Linear programming (LP; also called linear optimization) is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (mathematical optimization). More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polyhedron, which is a set defined as the intersection of finitely many half spaces, each of which is defined by a linear inequality. Its objective function is a real-valued affine function defined on this polyhedron. A linear programming algorithm finds a point in the polyhedron where this function has the smallest (or largest) value if such a point exists.
Linear Quadratic Estimation
Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, containing noise (random variations) and other inaccuracies, and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone. More formally, the Kalman filter operates recursively on streams of noisy input data to produce a statistically optimal estimate of the underlying system state. The filter is named after Rudolf (Rudy) E. Kálmán, one of the primary developers of its theory. The Kalman filter has numerous applications in technology. A common application is for guidance, navigation and control of vehicles, particularly aircraft and spacecraft. Furthermore, the Kalman filter is a widely applied concept in time series analysis used in fields such as signal processing and econometrics. Kalman filters also are one of the main topics in the field of Robotic motion planning and control, and sometimes included in Trajectory optimization.
Linear Regression In statistics, linear regression is an approach for modeling the relationship between a scalar dependent variable y and one or more explanatory variables (or independent variable) denoted X. The case of one explanatory variable is called simple linear regression. For more than one explanatory variable, the process is called multiple linear regression. (This term should be distinguished from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.)
In linear regression, data are modeled using linear predictor functions, and unknown model parameters are estimated from the data. Such models are called linear models. Most commonly, linear regression refers to a model in which the conditional mean of y given the value of X is an affine function of X. Less commonly, linear regression could refer to a model in which the median, or some other quantile of the conditional distribution of y given X is expressed as a linear function of X. Like all forms of regression analysis, linear regression focuses on the conditional probability distribution of y given X, rather than on the joint probability distribution of y and X, which is the domain of multivariate analysis.
Linear regression was the first type of regression analysis to be studied rigorously, and to be used extensively in practical applications. This is because models which depend linearly on their unknown parameters are easier to fit than models which are non-linearly related to their parameters and because the statistical properties of the resulting estimators are easier to determine.
Linear regression has many practical uses. Most applications fall into one of the following two broad categories:
• If the goal is prediction, or forecasting, or reduction, linear regression can be used to fit a predictive model to an observed data set of y and X values. After developing such a model, if an additional value of X is then given without its accompanying value of y, the fitted model can be used to make a prediction of the value of y.
• Given a variable y and a number of variables X1, …, Xp that may be related to y, linear regression analysis can be applied to quantify the strength of the relationship between y and the Xj, to assess which Xj may have no relationship with y at all, and to identify which subsets of the Xj contain redundant information about y.
Linear regression models are often fitted using the least squares approach, but they may also be fitted in other ways, such as by minimizing the ‘lack of fit’ in some other norm (as with least absolute deviations regression), or by minimizing a penalized version of the least squares loss function as in ridge regression (L2-norm penalty) and lasso (L1-norm penalty). Conversely, the least squares approach can be used to fit models that are not linear models. Thus, although the terms ‘least squares’ and ‘linear model’ are closely linked, they are not synonymous.
Linear Superiorization
Linear superiorization (abbreviated: LinSup) considers linear programming (LP) problems wherein the constraints as well as the objective function are linear. It allows to steer the iterates of a feasibility-seeking iterative process toward feasible points that have lower (not necessarily minimal) values of the objective function than points that would have been reached by the same feasiblity-seeking iterative process without superiorization. Using a feasibility-seeking iterative process that converges even if the linear feasible set is empty, LinSup generates an iterative sequence that converges to a point that minimizes a proximity function which measures the linear constraints violation. In addition, due to LinSup’s repeated objective function reduction steps such a point will most probably have a reduced objective function value. We present an exploratory experimental result that illustrates the behavior of LinSup on an infeasible LP problem.
Linear Unified LASSO
We propose a rescaled LASSO, by premultipying the LASSO with a matrix term, namely linear unified LASSO (LLASSO) for multicollinear situations. Our numerical study has shown that the LLASSO is comparable with other sparse modeling techniques and often outperforms the LASSO and elastic net. Our findings open new visions about using the LASSO still for sparse modeling and variable selection. We conclude our study by pointing that the LLASSO can be solved by the same efficient algorithm for solving the LASSO and suggest to follow the same construction technique for other penalized estimators.
Linear-Time Clustering Algorithm
In this paper, we first propose a new iterative algorithm, called the K-sets+ algorithm for clustering data points in a semi-metric space, where the distance measure does not necessarily satisfy the triangular inequality. We show that the K-sets+ algorithm converges in a finite number of iterations and it retains the same performance guarantee as the K-sets algorithm for clustering data points in a metric space. We then extend the applicability of the K-sets+ algorithm from data points in a semi-metric space to data points that only have a symmetric similarity measure. Such an extension leads to great reduction of computational complexity. In particular, for an n * n similarity matrix with m nonzero elements in the matrix, the computational complexity of the K-sets+ algorithm is O((Kn + m)I), where I is the number of iterations. The memory complexity to achieve that computational complexity is O(Kn + m). As such, both the computational complexity and the memory complexity are linear in n when the n * n similarity matrix is sparse, i.e., m = O(n). We also conduct various experiments to show the effectiveness of the K-sets+ algorithm by using a synthetic dataset from the stochastic block model and a real network from the WonderNetwork website.
Linear-Time Detection of Non-Linear Changes
Change detection in multivariate time series has applications in many domains, including health care and network monitoring. A common approach to detect changes is to compare the divergence between the distributions of a reference window and a test window. When the number of dimensions is very large, however, the na¨ıve approach has both quality and efficiency issues: to ensure robustness the window size needs to be large, which not only leads to missed alarms but also increases runtime. To this end, we propose LIGHT, a linear-time algorithm for robustly detecting non-linear changes in massively high dimensional time series. Importantly, LIGHT provides high flexibility in choosing the window size, allowing the domain expert to fit the level of details required. To do such, we 1) perform scalable PCA to reduce dimensionality, 2) perform scalable factorization of the joint distribution, and 3) scalably compute divergences between these lower dimensional distributions. Extensive empirical evaluation on both synthetic and real-world data show that LIGHT outperforms state of the art with up to 100% improvement in both quality and efficiency.
Linguistic Descriptions of Complex Phenomena
Linguistic Descriptions of Complex Phenomena (LDCP) is an architecture and methodology that allows us to model complex phenomena, interpreting input data, and generating automatic text reports customized to the user needs (see <doi:10.1016/j.ins.2016.11.002> and <doi:10.1007/s00500-016-2430-5> ).
Link Function In GLM, the link function provides the relationship between the linear predictor and the mean of the distribution function. There are many commonly used link functions, and their choice can be somewhat arbitrary. It makes sense to try to match the domain of the link function to the range of the distribution function’s mean.
Link Prediction Given a snapshot of a social network, can we infer which new interactions among its members are likely to occur in the near future? We formalize this question as the link prediction problem, and develop approaches to link prediction based on measures for analyzing the \proximity” of nodes in a network. Experiments on large co-authorship networks suggest that information about future interactions can be extracted from network topology alone, and that fairly subtle measures for detecting node proximity can outperform more direct measures.
Linked Data Ranking Algorithm
The advances of the Linked Open Data (LOD) initiative are giving rise to a more structured Web of data. Indeed, a few datasets act as hubs (e.g., DBpedia) connecting many other datasets. They also made possible new Web services for entity detection inside plain text (e.g., DBpedia Spotlight), thus allowing for new applications that can benefit from a combination of the Web of documents and the Web of data. To ease the emergence of these new applications, we propose a query-biased algorithm (LDRANK) for the ranking of web of data resources with associated textual data. Our algorithm combines link analysis with dimensionality reduction. We use crowdsourcing for building a publicly available and reusable dataset for the evaluation of query-biased ranking of Web of data resources detected in Web pages. We show that, on this dataset, LDRANK outperforms the state of the art. Finally, we use this algorithm for the construction of semantic snippets of which we evaluate the usefulness with a crowdsourcing-based approach.
Linked Matrix Factorization
In recent years, a number of methods have been developed for the dimension reduction and decomposition of multiple linked high-content data matrices. Typically these methods assume that just one dimension, rows or columns, is shared among the data sources. This shared dimension may represent common features that are measured for different sample sets (i.e., horizontal integration) or a common set of samples with measurements for different feature sets (i.e., vertical integration). In this article we introduce an approach for simultaneous horizontal and vertical integration, termed Linked Matrix Factorization (LMF), for the more general situation where some matrices share rows (e.g., features) and some share columns (e.g., samples). Our motivating application is a cytotoxicity study with accompanying genomic and molecular chemical attribute data. In this data set, the toxicity matrix (cell lines $\times$ chemicals) shares its sample set with a genotype matrix (cell lines $\times$ SNPs), and shares its feature set with a chemical molecular attribute matrix (chemicals $\times$ attributes). LMF gives a unified low-rank factorization of these three matrices, which allows for the decomposition of systematic variation that is shared among the three matrices and systematic variation that is specific to each matrix. This may be used for efficient dimension reduction, exploratory visualization, and the imputation of missing data even when entire rows or columns are missing from a constituent data matrix. We present theoretical results concerning the uniqueness, identifiability, and minimal parametrization of LMF, and evaluate it with extensive simulation studies.
Linked Micromaps Linked Micromaps is a graphing program written in Java. It allows users to view multiple variables interactively and compare statistics across regions (states, counties, registries, hospitals) as well as across time. It supports six types of graph:
• bar graphs;
• box plots;
• raw data tables;
• point graphs;
• point graphs with arrow; and
• point graphs with confidence intervals.
Liquid Analytics Liquid analytics. That’s the part that automatically updates and refines the training sets, rules, inferences, confidence intervals and predictions, every day, as mutating data keeps pouring non-stop in the databases (be it NoSQL or not). While this (most of the time) still ends up being coded in production mode by software engineers or developers, the framework and logical architecture is designed by data scientists. Because of this, data science is to data floods what statistical science is to frozen data.
Literate Programming Literate programming is an approach to programming introduced by Donald Knuth in which a program is given as an explanation of the program logic in a natural language, such as English, interspersed with snippets of macros and traditional source code, from which a compilable source code can be generated.
littler (“little R”) littler provides the r program, a simplified command-line interface for GNU R. This allows direct execution of commands, use in piping where the output of one program supplies the input of the next, as well as adding the ability for writing hash-bang scripts, i.e. creating executable files starting with, say, #!/usr/bin/r.
GNU R, a language and environment for statistical computing and graphics, provides a wonderful system for ‘programming with data’ as well as interactive exploratory analysis, often involving graphs. Sometimes, however, simple scripts are desired. While R can be used in batch mode, and while so-called here documents can be crafted, a long-standing need for a scripting front-end has often been expressed by the R Community. littler (pronounced little R and written r) aims to fill this need.
Ljung-Box Test The Ljung-Box test (named for Greta M. Ljung and George E. P. Box) is a type of statistical test of whether any of a group of autocorrelations of a time series are different from zero. Instead of testing randomness at each distinct lag, it tests the ‘overall’ randomness based on a number of lags, and is therefore a portmanteau test. This test is sometimes known as the Ljung-Box Q test, and it is closely connected to the Box-Pierce test (which is named after George E. P. Box and David A. Pierce). In fact, the Ljung-Box test statistic was described explicitly in the paper that led to the use of the Box-Pierce statistic, and from which that statistic takes its name. The Box-Pierce test statistic is a simplified version of the Ljung-Box statistic for which subsequent simulation studies have shown poor performance. The Ljung-Box test is widely applied in econometrics and other applications of time series analysis.
Lloyd-Max In computer science and electrical engineering, Lloyd’s algorithm, also known as Voronoi iteration or relaxation, is an algorithm named after Stuart P. Lloyd for finding evenly spaced sets of points in subsets of Euclidean spaces, and partitions of these subsets into well-shaped and uniformly sized convex cells. Like the closely related k-means clustering algorithm, it repeatedly finds the centroid of each set in the partition, and then re-partitions the input according to which of these centroids is closest. However, Lloyd’s algorithm differs from k-means clustering in that its input is a continuous geometric region rather than a discrete set of points. Thus, when re-partitioning the input, Lloyd’s algorithm uses Voronoi diagrams rather than simply determining the nearest center to each of a finite set of points as the k-means algorithm does. Although the algorithm may be applied most directly to the Euclidean plane, similar algorithms may also be applied to higher-dimensional spaces or to spaces with other non-Euclidean metrics. Lloyd’s algorithm can be used to construct close approximations to centroidal Voronoi tessellations of the input, which can be used for quantization, dithering, and stippling. Other applications of Lloyd’s algorithm include smoothing of triangle meshes in the finite element method.
“Compressive K-means”
Local Average Treatment Effect
Local Binary Convolution
We propose local binary convolution (LBC), an efficient alternative to convolutional layers in standard convolutional neural networks (CNN). The design principles of LBC are motivated by local binary patterns (LBP). The LBC layer comprises of a set of fixed sparse pre-defined binary convolutional filters that are not updated during the training process, a non-linear activation function and a set of learnable linear weights. The linear weights combine the activated filter responses to approximate the corresponding activated filter responses of a standard convolutional layer. The LBC layer affords significant parameter savings, 9x to 169x in the number of learnable parameters compared to a standard convolutional layer. Furthermore, due to lower model complexity and sparse and binary nature of the weights also results in up to 9x to 169x savings in model size compared to a standard convolutional layer. We demonstrate both theoretically and experimentally that our local binary convolution layer is a good approximation of a standard convolutional layer. Empirically, CNNs with LBC layers, called local binary convolutional neural networks (LBCNN), reach state-of-the-art performance on a range of visual datasets (MNIST, SVHN, CIFAR-10, and a subset of ImageNet) while enjoying significant computational savings.
Local Error Driven and Associative Biologically Realistic Algorithm
The algorithm Leabra (local error driven and associative biologically realistic algorithm) allows for the construction of artificial neural networks that are biologically realistic and balance supervised and unsupervised learning within a single framework.
Local Expansion via Minimum One Norm
We propose a novel approach for finding overlapping communities called LEMON (Local Expansion via Minimum One Norm). The algorithm finds the community by seeking a sparse vector in the span of the local spectra such that the seeds are in its support. We show that LEMON can achieve the highest detection accuracy among state-of-the-art proposals. The running time depends on the size of the community rather than that of the entire graph. The algorithm is easy to implement, and is highly parallelizable.
Local False Discovery Rate
“False Discovery Rate”
Local Fisher Discriminant Analysis
Local Interpretable Model-Agnostic Explanations
Machine learning is at the core of many recent advances in science and technology. With computers beating professionals in games like Go, many people have started asking if machines would also make for better drivers or even better doctors. In many applications of machine learning, users are asked to trust a model to help them make decisions. A doctor will certainly not operate on a patient simply because “the model said so.” Even in lower-stakes situations, such as when choosing a movie to watch from Netflix, a certain measure of trust is required before we surrender hours of our time based on a model. Despite the fact that many machine learning models are black boxes, understanding the rationale behind the model’s predictions would certainly help users decide when to trust or not to trust their predictions. An example is shown in Figure 1, in which a model predicts that a certain patient has the flu. The prediction is then explained by an ‘explainer’ that highlights the symptoms that are most important to the model. With this information about the rationale behind the model, the doctor is now empowered to trust the model—or not.
Local Outlier Factor
In anomaly detection, the local outlier factor (LOF) is an algorithm proposed by Markus M. Breunig, Hans-Peter Kriegel, Raymond T. Ng and Jörg Sander in 2000 for finding anomalous data points by measuring the local deviation of a given data point with respect to its neighbours. LOF shares some concepts with DBSCAN and OPTICS such as the concepts of ‘core distance’ and ‘reachability distance’, which are used for local density estimation.
Local Projections In this paper, we propose a novel approach for outlier detection, called local projections, which is based on concepts of Local Outlier Factor (LOF) (Breunig et al., 2000) and RobPCA (Hubert et al., 2005). By using aspects of both methods, our algorithm is robust towards noise variables and is capable of performing outlier detection in multi-group situations. We are further not reliant on a specific underlying data distribution. For each observation of a dataset, we identify a local group of dense nearby observations, which we call a core, based on a modification of the k-nearest neighbours algorithm. By projecting the dataset onto the space spanned by those observations, two aspects are revealed. First, we can analyze the distance from an observation to the center of the core within the projection space in order to provide a measure of quality of description of the observation by the projection. Second, we consider the distance of the observation to the projection space in order to assess the suitability of the core for describing the outlyingness of the observation. These novel interpretations lead to a univariate measure of outlyingness based on aggregations over all local projections, which outperforms LOF and RobPCA as well as other popular methods like PCOut (Filzmoser et al., 2008) and subspace-based outlier detection (Kriegel et al., 2009) in our simulation setups. Experiments in the context of real-word applications employing datasets of various dimensionality demonstrate the advantages of local projections.
Local Regression
LOESS and LOWESS (locally weighted scatterplot smoothing) are two strongly related non-parametric regression methods that combine multiple regression models in a k-nearest-neighbor-based meta-model. “LOESS” is a later generalization of LOWESS; although it is not a true initialism, it may be understood as standing for “LOcal regrESSion”.
Local Reparameterization Network
Recent breakthroughs in computer vision make use of large deep neural networks, utilizing the substantial speedup offered by GPUs. For applications running on limited hardware however, high precision real-time processing can still be a challenge. One approach to solve this problem is learning networks with binary or ternary weights, thus removing the need to calculate multiplications and significantly reduce memory size and access. In this work we introduce LR-nets (Local reparameterization networks), a new method for training neural networks with discrete weights using stochastic parameters. We show how a simple modification to the local reparameterization trick, previously used to train Gaussian distributed weights, allows us to train discrete weights. We tested our method on MNIST, CIFAR-10 and ImageNet, achieving state-of-the-art results compared to previous binary and ternary models.
Local Shrunk Discriminant Analysis
Dimensionality reduction is a crucial step for pattern recognition and data mining tasks to overcome the curse of dimensionality. Principal component analysis (PCA) is a traditional technique for unsupervised dimensionality reduction, which is often employed to seek a projection to best represent the data in a least-squares sense, but if the original data is nonlinear structure, the performance of PCA will quickly drop. An supervised dimensionality reduction algorithm called Linear discriminant analysis (LDA) seeks for an embedding transformation, which can work well with Gaussian distribution data or single-modal data, but for non-Gaussian distribution data or multimodal data, it gives undesired results. What is worse, the dimension of LDA cannot be more than the number of classes. In order to solve these issues, Local shrunk discriminant analysis (LSDA) is proposed in this work to process the non-Gaussian distribution data or multimodal data, which not only incorporate both the linear and nonlinear structures of original data, but also learn the pattern shrinking to make the data more flexible to fit the manifold structure. Further, LSDA has more strong generalization performance, whose objective function will become local LDA and traditional LDA when different extreme parameters are utilized respectively. What is more, a new efficient optimization algorithm is introduced to solve the non-convex objective function with low computational cost. Compared with other related approaches, such as PCA, LDA and local LDA, the proposed method can derive a subspace which is more suitable for non-Gaussian distribution and real data. Promising experimental results on different kinds of data sets demonstrate the effectiveness of the proposed approach.
Locality Sensitive Hashing
Locality-sensitive hashing (LSH) is a method of performing probabilistic dimension reduction of high-dimensional data. The basic idea is to hash the input items so that similar items are mapped to the same buckets with high probability (the number of buckets being much smaller than the universe of possible input items). This is different from the conventional hash functions, such as those used in cryptography, as in the LSH case the goal is to maximize probability of ‘collision’ of similar items rather than avoid collisions. Note how locality-sensitive hashing, in many ways, mirrors data clustering and Nearest neighbor search.
Locally Estimated Scatterplot Smoothing
LOESS and LOWESS (locally weighted scatterplot smoothing) are two strongly related non-parametric regression methods that combine multiple regression models in a k-nearest-neighbor-based meta-model. “LOESS” is a later generalization of LOWESS; although it is not a true initialism, it may be understood as standing for “LOcal regrESSion”.
Locally Linear Embedding
Locally Linear Embedding (LLE), an unsupervised learning algorithm that computes low dimensional, neighborhood preserving embeddings of high dimensional data. LLE attempts to discover nonlinear structure in high dimensional data by exploiting the local symmetries of linear reconstructions. Notably, LLE maps its inputs into a single global coordinate system of lower dimensionality, and its optimizations – though capable of generating highly nonlinear embeddings – do not involve local minima.
Locally Smoothed Neural Network
Convolutional Neural Networks (CNN) and the locally connected layer are limited in capturing the importance and relations of different local receptive fields, which are often crucial for tasks such as face verification, visual question answering, and word sequence prediction. To tackle the issue, we propose a novel locally smoothed neural network (LSNN) in this paper. The main idea is to represent the weight matrix of the locally connected layer as the product of the kernel and the smoother, where the kernel is shared over different local receptive fields, and the smoother is for determining the importance and relations of different local receptive fields. Specifically, a multi-variate Gaussian function is utilized to generate the smoother, for modeling the location relations among different local receptive fields. Furthermore, the content information can also be leveraged by setting the mean and precision of the Gaussian function according to the content. Experiments on some variant of MNIST clearly show our advantages over CNN and locally connected layer.
In this paper we present LocLinkVis (Locate-Link-Visualize); a system which supports exploratory information access to a document collection based on geo-referencing and visualization. It uses a gazetteer which contains representations of places ranging from countries to buildings, and that is used to recognize toponyms, disambiguate them into places, and to visualize the resulting spatial footprints.
Location Determination Problem
Logistic Regression In statistics, logistic regression, or logit regression, is a type of probabilistic statistical classification model. It is also used to predict a binary response from a binary predictor, used for predicting the outcome of a categorical dependent variable (i.e., a class label) based on one or more predictor variables (features). That is, it is used in estimating empirical values of the parameters in a qualitative response model. The probabilities describing the possible outcomes of a single trial are modeled, as a function of the explanatory (predictor) variables, using a logistic function. Frequently (and subsequently in this article) “logistic regression” is used to refer specifically to the problem in which the dependent variable is binary-that is, the number of available categories is two-while problems with more than two categories are referred to as multinomial logistic regression or, if the multiple categories are ordered, as ordered logistic regression.
logit The logit function is the inverse of the sigmoidal “logistic” function used in mathematics, especially in statistics. When the function’s parameter represents a probability p, the logit function gives the log-odds, or the logarithm of the odds p/(1-p).
LogitBoost Autoregressive Networks Multivariate binary distributions can be decomposed into products of univariate conditional distributions. Recently popular approaches have modeled these conditionals through neural networks with sophisticated weight-sharing structures. It is shown that state-of-the-art performance on several standard benchmark datasets can actually be achieved by training separate probability estimators for each dimension. In that case, model training can be trivially parallelized over data dimensions. On the other hand, complexity control has to be performed for each learned conditional distribution. Three possible methods are considered and experimentally compared. The estimator that is employed for each conditional is LogitBoost. Similarities and differences between the proposed approach and autoregressive models based on neural networks are discussed in detail.
Log-Likelihood For many applications, the natural logarithm of the likelihood function, called the log-likelihood, is more convenient to work with. Because the logarithm is a monotonically increasing function, the logarithm of a function achieves its maximum value at the same points as the function itself, and hence the log-likelihood can be used in place of the likelihood in maximum likelihood estimation and related techniques. Finding the maximum of a function often involves taking the derivative of a function and solving for the parameter being maximized, and this is often easier when the function being maximized is a log-likelihood rather than the original likelihood function. For example, some likelihood functions are for the parameters that explain a collection of statistically independent observations. In such a situation, the likelihood function factors into a product of individual likelihood functions. The logarithm of this product is a sum of individual logarithms, and the derivative of a sum of terms is often easier to compute than the derivative of a product. In addition, several common distributions have likelihood functions that contain products of factors involving exponentiation. The logarithm of such a function is a sum of products, again easier to differentiate than the original function. In phylogenetics the log-likelihood ratio is sometimes termed support and the log-likelihood function support function. However, given the potential for confusion with the mathematical meaning of ‘support’ this terminology is rarely used outside this field.
Log-Linear Model A log-linear model is a mathematical model that takes the form of a function whose logarithm is a first-degree polynomial function of the parameters of the model, which makes it possible to apply (possibly multivariate) linear regression.
Log-rank Test In statistics, the log-rank test is a hypothesis test to compare the survival distributions of two samples. It is a nonparametric test and appropriate to use when the data are right skewed and censored (technically, the censoring must be non-informative). It is widely used in clinical trials to establish the efficacy of a new treatment in comparison with a control treatment when the measurement is the time to event (such as the time from initial treatment to a heart attack). The test is sometimes called the Mantel-Cox test, named after Nathan Mantel and David Cox. The log-rank test can also be viewed as a time-stratified Cochran-Mantel-Haenszel test.
LoIDE Logic-based paradigms are nowadays widely used in many different fields, also thank to the availability of robust tools and systems that allow the development of real-world and industrial applications. In this work we present LoIDE, an advanced and modular web-editor for logic-based languages that also integrates with state-of-the-art solvers.
Long- and Short-Term Time-Series Network
Multivariate time series forecasting is an important machine learning problem across many domains, including predictions of solar plant energy output, electricity consumption, and traffic jam situation. Temporal data arise in these real-world applications often involves a mixture of long-term and short-term patterns, for which traditional approaches such as Autoregressive models and Gaussian Process may fail. In this paper, we proposed a novel deep learning framework, namely Long- and Short-term Time-series network (LSTNet), to address this open challenge. LSTNet uses the Convolution Neural Network (CNN) to extract short-term local dependency patterns among variables, and the Recurrent Neural Network (RNN) to discover long-term patterns and trends. In our evaluation on real-world data with complex mixtures of repetitive patterns, LSTNet achieved significant performance improvements over that of several state-of-the-art baseline methods.
Long Short Term Memory
Long short term memory (LSTM) is a recurrent neural network (RNN) architecture (an artificial neural network) published in 1997 by Sepp Hochreiter and Jürgen Schmidhuber. Like most RNNs, an LSTM network is universal in the sense that given enough network units it can compute anything a conventional computer can compute, provided it has the proper weight matrix, which may be viewed as its program. (Of course, finding such a weight matrix is more challenging with some problems than with others.) Unlike traditional RNNs, an LSTM network is well-suited to learn from experience to classify, process and predict time series when there are very long time lags of unknown size between important events. This is one of the main reasons why LSTM outperforms alternative RNNs and Hidden Markov Models and other sequence learning methods in numerous applications. For example, LSTM achieved the best known results in unsegmented connected handwriting recognition, and in 2009 won the ICDAR handwriting competition. LSTM networks have also been used for automatic speech recognition, and were a major component of a network that recently achieved a record 17.7% phoneme error rate on the classic TIMIT natural speech dataset.
Longitudinal Study A longitudinal survey is a correlational research study that involves repeated observations of the same variables over long periods of time — often many decades. It is a type of observational study. Longitudinal studies are often used in psychology to study developmental trends across the life span, and in sociology to study life events throughout lifetimes or generations. The reason for this is that, unlike cross-sectional studies, in which different individuals with same characteristics are compared, longitudinal studies track the same people, and therefore the differences observed in those people are less likely to be the result of cultural differences across generations. Because of this benefit, longitudinal studies make observing changes more accurate, and they are applied in various other fields. In medicine, the design is used to uncover predictors of certain diseases. In advertising, the design is used to identify the changes that advertising has produced in the attitudes and behaviors of those within the target audience who have seen the advertising campaign. Because most longitudinal studies are observational, in the sense that they observe the state of the world without manipulating it, it has been argued that they may have less power to detect causal relationships than experiments. But because of the repeated observation at the individual level, they have more power than cross-sectional observational studies, by virtue of being able to exclude time-invariant unobserved individual differences, and by virtue of observing the temporal order of events. Some of the disadvantages of longitudinal study include the fact that they take a lot of time and are very expensive. Therefore, they are not very convenient. Longitudinal studies allow social scientists to distinguish short from long-term phenomena, such as poverty. If the poverty rate is 10% at a point in time, this may mean that 10% of the population are always poor, or that the whole population experiences poverty for 10% of the time. It is impossible to conclude which of these possibilities is the case using one-off cross-sectional studies. Types of longitudinal studies include cohort studies and panel studies. Cohort studies sample a cohort, defined as a group experiencing some event (typically birth) in a selected time period, and studying them at intervals through time. Panel studies sample a cross-section, and survey it at (usually regular) intervals. A retrospective study is a longitudinal study that looks back in time. For instance, a researcher may look up the medical records of previous years to look for a trend.
Long-Range Dependency
Long-range dependency (LRD), also called long memory or long-range persistence, is a phenomenon that may arise in the analysis of spatial or time series data. It relates to the rate of decay of statistical dependence, with the implication that this decays more slowly than an exponential decay, typically a power-like decay. LRD is often related to self-similar processes or fields. LRD has been used in various fields such as internet traffic modelling, econometrics, hydrology, linguistics and the earth sciences. Different mathematical definitions of LRD are used for different contexts and purposes.
Lookup-Based Convolutional Neural Network
Porting state of the art deep learning algorithms to resource constrained compute platforms (e.g. VR, AR, wearables) is extremely challenging. We propose a fast, compact, and accurate model for convolutional neural networks that enables efficient learning and inference. We introduce LCNN, a lookup-based convolutional neural network that encodes convolutions by few lookups to a dictionary that is trained to cover the space of weights in CNNs. Training LCNN involves jointly learning a dictionary and a small set of linear combinations. The size of the dictionary naturally traces a spectrum of trade-offs between efficiency and accuracy. Our experimental results on ImageNet challenge show that LCNN can offer 3.2x speedup while achieving 55.1% top-1 accuracy using AlexNet architecture. Our fastest LCNN offers 37.6x speed up over AlexNet while maintaining 44.3% top-1 accuracy. LCNN not only offers dramatic speed ups at inference, but it also enables efficient training. In this paper, we show the benefits of LCNN in few-shot learning and few-iteration learning, two crucial aspects of on-device training of deep learning models.
Lorenz Curve In economics, the Lorenz curve is a graphical representation of the cumulative distribution function of the empirical probability distribution of wealth or income, and was developed by Max O. Lorenz in 1905 for representing inequality of the wealth distribution. The curve is a graph showing the proportion of overall income or wealth assumed by the bottom x% of the people, although this is not rigorously true for a finite population (see below). It is often used to represent income distribution, where it shows for the bottom x% of households, what percentage (y%) of the total income they have. The percentage of households is plotted on the x-axis, the percentage of income on the y-axis. It can also be used to show distribution of assets. In such use, many economists consider it to be a measure of social inequality. The concept is useful in describing inequality among the size of individuals in ecology and in studies of biodiversity, where the cumulative proportion of species is plotted against the cumulative proportion of individuals. It is also useful in business modeling: e.g., in consumer finance, to measure the actual percentage y% of delinquencies attributable to the x% of people with worst risk scores.
Loss Distributional Approach
While AMA does not specify the use of any particular modeling technique, one common approach taken in the banking industry is the Loss Distribution Approach (LDA). With LDA, a bank first segments operational losses into homogeneous segments, called unit of measure’s (UoMs). For each unit of measure, the bank then constructs a loss distribution that represents its expectation of total losses that can materialize in a one-year horizon. Given that data sufficiency is a major challenge for the industry, annual loss distribution cannot be built directly using annual loss figures. Instead, a bank will develop a frequency distribution that describes the number of loss events in a given year, and a severity distribution that describes the loss amount of a single loss event. The frequency and severity distributions are assumed to be independent. The convolution of these two distributions then give rise to the (annual) loss distribution.
Loss Function In mathematical optimization, statistics, decision theory and machine learning, a loss function or cost function is a function that maps an event or values of one or more variables onto a real number intuitively representing some ‘cost’ associated with the event. An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its negative (sometimes called a reward function or a utility function), in which case it is to be maximized. In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an instance of data. The concept, as old as Laplace, was reintroduced in statistics by Abraham Wald in the middle of the 20th Century. In the context of economics, for example, this is usually economic cost or regret. In classification, it is the penalty for an incorrect classification of an example. In actuarial science, it is used in an insurance context to model benefits paid over premiums, particularly since the works of Harald Cramér in the 1920s. In optimal control the loss is the penalty for failing to achieve a desired value. In financial risk management the function is precisely mapped to a monetary loss.
Lotka’s Law Lotka’s law, named after Alfred J. Lotka, is one of a variety of special applications of Zipf’s law. It describes the frequency of publication by authors in any given field. It states that the number of authors making n contributions is about 1/n^{a} of those making one contribution, where a nearly always equals two. More plainly, the number of authors publishing a certain number of articles is a fixed ratio to the number of authors publishing a single article. As the number of articles published increases, authors producing that many publications become less frequent. There are 1/4 as many authors publishing two articles within a specified time period as there are single-publication authors, 1/9 as many publishing three articles, 1/16 as many publishing four articles, etc. Though the law itself covers many disciplines, the actual ratios involved (as a function of ‘a’) are very discipline-specific.
Louvain Method Our method, that we call Louvain Method (because, even though the co-authors now hold positions in Paris, London and Louvain, the method was devised when they all were in Louvain), outperforms other methods in terms of computation time, which allows us to analyze networks of unprecedented size (e.g. the analysis of a typical network of 2 million nodes only takes 2 minutes). The Louvain method has also been to shown to be very accurate by focusing on ad-hoc networks with known community structure. Moreover, due to its hierarchical structure, which is reminiscent of renormalization methods, it allows to look at communities at different resolutions.
“Community Detection”
Louvain Modularity The Louvain Method for community detection is a method to extract communities from large networks created by Vincent Blondel. The method is a greedy optimization method that appears to run in time O(n log n).
Low Complexity Neural Network
Modern neural network architectures for large-scale learning tasks have substantially higher model complexities, which makes understanding, visualizing and training these architectures difficult. Recent contributions to deep learning techniques have focused on architectural modifications to improve parameter efficiency and performance. In this paper, we derive a continuous and differentiable error functional for a neural network that minimizes its empirical error as well as a measure of the model complexity. The latter measure is obtained by deriving a differentiable upper bound on the Vapnik-Chervonenkis (VC) dimension of the classifier layer of a class of deep networks. Using standard backpropagation, we realize a training rule that tries to minimize the error on training samples, while improving generalization by keeping the model complexity low. We demonstrate the effectiveness of our formulation (the Low Complexity Neural Network – LCNN) across several deep learning algorithms, and a variety of large benchmark datasets. We show that hidden layer neurons in the resultant networks learn features that are crisp, and in the case of image datasets, quantitatively sharper. Our proposed approach yields benefits across a wide range of architectures, in comparison to and in conjunction with methods such as Dropout and Batch Normalization, and our results strongly suggest that deep learning techniques can benefit from model complexity control methods such as the LCNN learning rule.
Low Dimensional Manifold Regularized Neural Network
Deep neural networks have proved very successful on archetypal tasks for which large training sets are available, but when the training data are scarce, their performance suffers from overfitting. Many existing methods of reducing overfitting are data-independent, and their efficacy is often limited when the training set is very small. Data-dependent regularizations are mostly motivated by the observation that data of interest lie close to a manifold, which is typically hard to parametrize explicitly and often requires human input of tangent vectors. These methods typically only focus on the geometry of the input data, and do not necessarily encourage the networks to produce geometrically meaningful features. To resolve this, we propose a new framework, the Low-Dimensional-Manifold-regularized neural Network (LDMNet), which incorporates a feature regularization method that focuses on the geometry of both the input data and the output features. In LDMNet, we regularize the network by encouraging the combination of the input data and the output features to sample a collection of low dimensional manifolds, which are searched efficiently without explicit parametrization. To achieve this, we directly use the manifold dimension as a regularization term in a variational functional. The resulting Euler-Lagrange equation is a Laplace-Beltrami equation over a point cloud, which is solved by the point integral method without increasing the computational complexity. We demonstrate two benefits of LDMNet in the experiments. First, we show that LDMNet significantly outperforms widely-used network regularizers such as weight decay and DropOut. Second, we show that LDMNet can be designed to extract common features of an object imaged via different modalities, which proves to be very useful in real-world applications such as cross-spectral face recognition.
Lowest Posterior Loss
This paper defines intrinsic credible regions, a method to produce objective Bayesian credible regions which only depends on the assumed model and the available data. Lowest posterior loss (LPL) regions are defined as Bayesian credible regions which contain values of minimum posterior expected loss: they depend both on the loss function and on the prior specification. An invariant, information-theory based loss function, the intrinsic discrepancy is argued to be appropriate for scientific communication. Intrinsic credible regions are the lowest posterior loss regions with respect to the intrinsic discrepancy loss and the appropriate reference prior. The proposed procedure is completely general, and it is invariant under both reparametrization and marginalization. The exact derivation of intrinsic credible regions often requires numerical integration, but good analytical approximations are provided. Special attention is given to one-dimensional intrinsic credible intervals; their coverage properties show that they are always approximate (and sometimes exact) frequentist confidence intervals.
Lowest Posterior Loss Interval
The Lowest Posterior Loss (LPL) interval (Bernardo, 2005), or LPLI, is a probability interval based on intrinsic discrepancy loss between prior and posterior distributions. The expected posterior loss is the loss associated with using a particular value theta[i] in theta of the parameter as the unknown true value of theta (Bernardo, 2005). Parameter values with smaller expected posterior loss should always be preferred. The LPL interval includes a region in which all parameter values have smaller expected posterior loss than those outside the region. Although any loss function could be used, the loss function should be invariant under reparameterization. Any intrinsic loss function is invariant under reparameterization, but not necessarily invariant under one-to-one transformations of data x. When a loss function is also invariant under one-to-one transformations, it is usually also invariant when reduced to a sufficient statistic. Only an intrinsic loss function that is invariant when reduced to a sufficient statistic should be considered. The intrinsic discrepancy loss is easily a superior loss function to the overused quadratic loss function, and is more appropriate than other popular measures, such as Hellinger distance, Kullback-Leibler divergence (KLD), and Jeffreys logarithmic divergence. The intrinsic discrepancy loss is also an information-theory related divergence measure. Intrinsic discrepancy loss is a symmetric, non-negative loss function, and is a continuous, convex function. Intrinsic discrepancy loss was introduced by Bernardo and Rueda (2002) in a different context: hypothesis testing. Formally, it is: delta f(p[2],p[1]) = min[kappa(p[2] | p[1]), kappa(p[1] | p[2])] where delta is the discrepancy, kappa is the KLD, and p[1] and p[2] are the probability distributions. The intrinsic discrepancy loss is the loss function, and the expected posterior loss is the mean of the directed divergences. The LPL interval is also called an intrinsic credible interval or intrinsic probability interval, and the area inside the interval is often called an intrinsic credible region or intrinsic probability region. In practice, whether a reference prior or weakly informative prior (WIP) is used, the LPL interval is usually very close to the HPD interval, though the posterior losses may be noticeably different. If LPL used a zero-one loss function, then the HPD interval would be produced. An advantage of the LPL interval over HPD interval (see p.interval) is that the LPL interval is invariant to reparameterization. This is due to the invariant reparameterization property of reference priors. The quantile-based probability interval is also invariant to reparameterization. The LPL interval enjoys the same advantage as the HPD interval does over the quantile-based probability interval: it does not produce equal tails when inappropriate. Compared with probability intervals, the LPL interval is slightly less convenient to calculate. Although the prior distribution is specified within the Model specification function, the user must specify it for the LPL.interval function as well. A comparison of the quantile-based probability interval, HPD interval, and LPL interval is available here: http://…/credible.
Low-Rank Kernel Subspace Clustering Most state-of-the-art subspace clustering methods only work with linear (or affine) subspaces. In this paper, we present a kernel subspace clustering method that can handle non-linear models. While an arbitrary kernel can non-linearly map data into high-dimensional Hilbert feature space, the data in the resulting feature space are very unlikely to have the desired subspace structures. By contrast, we propose to learn a low-rank kernel mapping, with which the mapped data in feature space are not only low-rank but also self-expressive, such that the low-dimensional subspace structures are present and manifested in the high-dimensional feature space. We have evaluated the proposed method extensively on both motion segmentation and image clustering benchmarks, and obtained superior results, outperforming the kernel subspace clustering method that uses standard kernels~\cite{patel2014kernel} and other state-of-the-art linear subspace clustering methods.
Lp Space In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki 1987) they were first introduced by Frigyes Riesz (Riesz 1910). Lp spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Lebesgue spaces have applications in physics, statistics, finance, engineering, and other disciplines.
Lua Lua is a powerful, efficient, lightweight, embeddable scripting language. It supports procedural programming, object-oriented programming, functional programming, data-driven programming, and data description. Lua combines simple procedural syntax with powerful data description constructs based on associative arrays and extensible semantics. Lua is dynamically typed, runs by interpreting bytecode with a register-based virtual machine, and has automatic memory management with incremental garbage collection, making it ideal for configuration, scripting, and rapid prototyping.
Luigi Luigi is a Python module that helps you build complex pipelines of batch jobs. It handles dependency resolution, workflow management, visualization etc. It also comes with Hadoop support built in.
Luigi is an open source Python-based data framework for building data pipelines. Instead of using an XML/YAML configuration of some sort, all the jobs and their dependencies are written as Python programs. Because it’s Python, developers can backtrack to figure out exactly how data is processed.
The framework makes it easier to build large data pipelines, with built-in checkpointing, failure recovery, parallel execution, command line integration, etc. Since it’s a Python program, any Python library assets can be reused. The Luigi framework itself is a couple of thousand lines, so it’s also easy to understand the entire mechanism.
Facebook built a similar internal system called Dataswarm (Video), which allows developers to manage the entire data pipeline on Git + Python.
While Luigi was originally invented for Spotify’s internal needs, companies such as Foursquare, Stripe, and Asana are using it in production.
Lurking Variable Lurking variables represent hidden information, and preclude a full understanding of phenomena of interest. Detection is usually based on serendipity — visual detection of unexplained, systematic variation. However, these approaches are doomed to fail if the lurking variables do not vary.