Related papers: Asymptotics of Constrained Quantization for Compac…
The problem of quantization of measures looks for best approximations of probability measures on a metric space by discrete measures supported on $N$ points, where the error of approximation is measured with respect to the Wasserstein…
In this work, we extend the classical framework of quantization for Borel probability measures defined on normed spaces $\mathbb{R}^k$ by introducing and analyzing the notions of the $n$th constrained quantization error, constrained…
We investigate the minimal error in approximating a general probability measure $\mu$ on $\mathbb{R}^d$ by the uniform measure on a finite set with prescribed cardinality $n$. The error is measured in the $p$-Wasserstein distance. In…
We provide a complete picture of the upper quantization dimension in terms of the R\'enyi dimension by proving that the upper quantization dimension $\bar{D}_{r}(\nu)$ of order $r>0$ for an arbitrary compactly supported Borel probability…
In this paper we study constrained subspace approximation problem. Given a set of $n$ points $\{a_1,\ldots,a_n\}$ in $\mathbb{R}^d$, the goal of the {\em subspace approximation} problem is to find a $k$ dimensional subspace that best…
We study the problem of compression for the purpose of similarity identification, where similarity is measured by the mean square Euclidean distance between vectors. While the asymptotical fundamental limits of the problem - the minimal…
Consider a set P of N random points on the unit sphere of dimension $d-1$, and the symmetrized set S = P union (-P). The halving polyhedron of S is defined as the convex hull of the set of centroids of N distinct points in S. We prove that…
Constrained quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with a finite number of supporting points lying on a specific set. The specific set is known as the…
Given a compact metric space $X$, the collection of Borel probability measures on $X$ can be made into a compact metric space via the Kantorovich metric. We partially generalize this well known result to projection-valued measures. In…
We consider the general problem of matching a subspace to a signal in R^N that has been observed indirectly (compressed) through a random projection. We are interested in the case where the collection of K-dimensional subspaces is…
Let $\{f_i\}_{i=1}^N$ be a set of equi-contractive similitudes on $\mathbb{R}^1$ satisfying the finite-type condition. We study the asymptotic quantization error for self-similar measures $\mu$ associated with $\{f_i\}_{i=1}^N$ and a…
This paper provides new error bounds on "consistent" reconstruction methods for signals observed from quantized random projections. Those signal estimation techniques guarantee a perfect matching between the available quantized data and a…
This paper is concerned with the study of the consistency of a variational method for probability measure quantization, deterministically realized by means of a minimizing principle, balancing power repulsion and attraction potentials. The…
Quantized compressive sensing (QCS) deals with the problem of representing compressive signal measurements with finite precision representation, i.e., a mandatory process in any practical sensor design. To characterize the signal…
We present a procedure for quantizing complex projective spaces $\mathbb{CP}^{p,q}$, $q\ge 1$, as well as construct relevant star products on these spaces. The quantization is made unique with the demand that it preserves the full isometry…
This work proposes and analyzes a compressed sensing approach to polynomial approximation of complex-valued functions in high dimensions. Of particular interest is the setting where the target function is smooth, characterized by a rapidly…
In this paper we extend results taken from compressed sensing to recover Hilbert-space valued vectors. This is an important problem in parametric function approximation in particular when the number of parameters is high. By expanding our…
This paper presents a detailed study of constrained quantization for both finite and infinite discrete probability distributions supported on subsets of the real line. Under specific geometric constraints - namely, a semicircular arc and…
Optical systems which measure independent random projections of a scene according to compressed sensing (CS) theory face a myriad of practical challenges related to the size of the physical platform, photon efficiency, the need for high…
A new exact projective penalty method is proposed for the equivalent reduction of constrained optimization problems to nonsmooth unconstrained ones. In the method, the original objective function is extended to infeasible points by summing…