Intrinsic Dimension (ID)
In signal processing of multidimensional signals, for example in computer vision, the intrinsic dimension of the signal describes how many variables are needed to represent the signal. For a signal of N variables, its intrinsic dimension M satisfies 0 = M = N. Usually the intrinsic dimension of a signal relates to variables defined in a Cartesian coordinate system. In general, however, it is also possible to describe the concept for non-Cartesian coordinates, for example, using polar coordinates. …

Dense Transformer Networks
The key idea of current deep learning methods for dense prediction is to apply a model on a regular patch centered on each pixel to make pixel-wise predictions. These methods are limited in the sense that the patches are determined by network architecture instead of learned from data. In this work, we propose the dense transformer networks, which can learn the shapes and sizes of patches from data. The dense transformer networks employ an encoder-decoder architecture, and a pair of dense transformer modules are inserted into each of the encoder and decoder paths. The novelty of this work is that we provide technical solutions for learning the shapes and sizes of patches from data and efficiently restoring the spatial correspondence required for dense prediction. The proposed dense transformer modules are differentiable, thus the entire network can be trained. We apply the proposed networks on natural and biological image segmentation tasks and show superior performance is achieved in comparison to baseline methods. …

Graph Information Ratio
We introduce the notion of information ratio $\text{Ir}(H/G)$ between two (simple, undirected) graphs $G$ and $H$, defined as the supremum of ratios $k/n$ such that there exists a mapping between the strong products $G^k$ to $H^n$ that preserves non-adjacency. Operationally speaking, the information ratio is the maximal number of source symbols per channel use that can be reliably sent over a channel with a confusion graph $H$, where reliability is measured w.r.t. a source confusion graph $G$. Various results are provided, including in particular lower and upper bounds on $\text{Ir}(H/G)$ in terms of different graph properties, inequalities and identities for behavior under strong product and disjoint union, relations to graph cores, and notions of graph criticality. Informally speaking, $\text{Ir}(H/G)$ can be interpreted as a measure of similarity between $G$ and $H$. We make this notion precise by introducing the concept of information equivalence between graphs, a more quantitative version of homomorphic equivalence. We then describe a natural partial ordering over the space of information equivalence classes, and endow it with a suitable metric structure that is contractive under the strong product. Various examples and open problems are discussed. …