Fuzzy Cognitive Map
A Fuzzy cognitive map is a cognitive map within which the relations between the elements (e.g. concepts, events, project resources) of a “mental landscape” can be used to compute the “strength of impact” of these elements. The theory behind that computation is fuzzy logic. …
Discrete Dantzig Selector
We propose a new high-dimensional linear regression estimator: the Discrete Dantzig Selector, which minimizes the number of nonzero regression coefficients, subject to a budget on the maximal absolute correlation between the features and the residuals. We show that the estimator can be expressed as a solution to a Mixed Integer Linear Optimization (MILO) problem—a computationally tractable framework that enables the computation of provably optimal global solutions. Our approach has the appealing characteristic that even if we terminate the optimization problem at an early stage, it exits with a certificate of sub-optimality on the quality of the solution. We develop new discrete first order methods, motivated by recent algorithmic developments in first order continuous convex optimization, to obtain high quality feasible solutions for the Discrete Dantzig Selector problem. Our proposal leads to advantages over the off-the-shelf state-of-the-art integer programming algorithms, which include superior upper bounds obtained for a given computational budget. When a solution obtained from the discrete first order methods is passed as a warm-start to a MILO solver, the performance of the latter improves significantly. Exploiting problem specific information, we propose enhanced MILO formulations that further improve the algorithmic performance of the MILO solvers. We demonstrate, both theoretically and empirically, that, in a wide range of regimes, the statistical properties of the Discrete Dantzig Selector are superior to those of popular $\ell_{1}$-based approaches. For problem instances with $p \approx 2500$ features and $n \approx 900$ observations, our computational framework delivers optimal solutions in a few minutes and certifies optimality within an hour. …
Robust Principal Component Analysis (ROBPCA)
We introduce a new method for robust principal component analysis (PCA). Classical PCA is based on the empirical covariance matrix of the data and hence is highly sensitive to outlying observations. Two robust approaches have been developed to date. The first approach is based on the eigenvectors of a robust scatter matrix such as the minimum covariance determinant or an S-estimator and is limited to relatively low-dimensional data. The second approach is based on projection pursuit and can handle highdimensional data. Here we propose the ROBPCA approach, which combines projection pursuit ideas with robust scatter matrix estimation. ROBPCA yields more accurate estimates at noncontaminated datasets and more robust estimates at contaminated data. ROBPCA can be computed rapidly, and is able to detect exact-fit situations. As a by-product, ROBPCA produces a diagnostic plot that displays and classifies the outliers. We apply the algorithm to several datasets from chemometrics and engineering. …