Related papers: Self Organizing Map algorithm and distortion measu…
Recent work of the author established dual representation theorems for certain vector spaces that arise in an important article of Allcock and Vaaler. These results constructed an object called a consistent map which acts like a measure on…
Controlling the internal representation space of a neural network is a desirable feature because it allows to generate new data in a supervised manner. In this paper we will show how this can be achieved while building a low-dimensional…
We study the problem of supervised learning a metric space under discriminative constraints. Given a universe $X$ and sets ${\cal S}, {\cal D}\subset {X \choose 2}$ of similar and dissimilar pairs, we seek to find a mapping $f:X\to Y$, into…
We consider the matching problem in the metric distortion framework. There are $n$ agents and $n$ items occupying points in a shared metric space, and the goal is to design a matching mechanism that outputs a low-cost matching between the…
Minimizing a convex function of a measure with a sparsity-inducing penalty is a typical problem arising, e.g., in sparse spikes deconvolution or two-layer neural networks training. We show that this problem can be solved by discretizing the…
The simultaneous orthogonal matching pursuit (SOMP) algorithm aims to find the joint support of a set of sparse signals acquired under a multiple measurement vector model. Critically, the analysis of SOMP depends on the maximal inner…
We study the optimal transport problem for $d>2$ discrete measures. This is a linear programming problem on $d$-tensors. It gives a way to compute a "distance" between two sets of discrete measures. We introduce an entropic regularization…
The Self-Organizing Map (SOM) with its related extensions is the most popular artificial neural algorithm for use in unsupervised learning, clustering, classification and data visualization. Over 5,000 publications have been reported in the…
In this paper, we study a connection between disintegration of measures and geometric properties of probability spaces. We prove a disintegration theorem, addressing disintegration from the perspective of an optimal transport problem. We…
Imitation learning algorithms have been interpreted as variants of divergence minimization problems. The ability to compare occupancy measures between experts and learners is crucial in their effectiveness in learning from demonstrations.…
A self-organizing map (SOM) is a type of competitive artificial neural network, which projects the high-dimensional input space of the training samples into a low-dimensional space with the topology relations preserved. This makes SOMs…
We explore the interplay between different definitions of distortion for mappings $f\colon X\to \mathbb{R}^2$, where $X$ is any metric surface, meaning that $X$ is homeomorphic to a domain in $\mathbb{R}^2$ and has locally finite…
We consider the problem of estimating the locations of a set of points in a k-dimensional euclidean space given a subset of the pairwise distance measurements between the points. We focus on the case when some fraction of these measurements…
We investigate the semi-discrete Optimal Transport (OT) problem, where a continuous source measure $\mu$ is transported to a discrete target measure $\nu$, with particular attention to the OT map approximation. In this setting, Stochastic…
Pocock and Simon's minimization method is a popular approach for covariate-adaptive randomization in clinical trials. Valid statistical inference with data collected under the minimization method requires the knowledge of the limiting…
Image feature classification is a challenging problem in many computer vision applications, specifically, in the fields of remote sensing, image analysis and pattern recognition. In this paper, a novel Self Organizing Map, termed improved…
In a setting where $m$ items need to be partitioned among $n$ agents, we evaluate the performance of mechanisms that take as input each agent's \emph{ordinal preferences}, i.e., their ranking of the items from most- to least-preferred. The…
Optimal transport induces the Earth Mover's (Wasserstein) distance between probability distributions, a geometric divergence that is relevant to a wide range of problems. Over the last decade, two relaxations of optimal transport have been…
We study independent and identically distributed random iterations of continuous maps defined on a connected closed subset $S$ of the Euclidean space $\mathbb{R}^{k}$. We assume the maps are monotone (with respect to a suitable partial…
We consider nonconforming methods for symmetric elliptic problems and characterize their quasi-optimality in terms of suitable notions of stability and consistency. The quasi-optimality constant is determined and the possible impact of…