Structured Matrix Recovery via the Generalized Dantzig Selector

TL;DR

This paper develops a theoretical framework for structured matrix recovery using the generalized Dantzig selector, extending analysis beyond low-rank matrices to more general structures with practical applications.

Contribution

It introduces a non-asymptotic analysis method for estimating structured matrices via the generalized Dantzig selector, applicable to a broad class of matrix structures.

Findings

Estimation error bounds depend on geometric measures of the matrix structure

Derived bounds for unitarily invariant norms covering most practical matrix norms

Examples demonstrate the applicability of the theoretical results

Abstract

In recent years, structured matrix recovery problems have gained considerable attention for its real world applications, such as recommender systems and computer vision. Much of the existing work has focused on matrices with low-rank structure, and limited progress has been made matrices with other types of structure. In this paper we present non-asymptotic analysis for estimation of generally structured matrices via the generalized Dantzig selector under generic sub-Gaussian measurements. We show that the estimation error can always be succinctly expressed in terms of a few geometric measures of suitable sets which only depend on the structure of the underlying true matrix. In addition, we derive the general bounds on these geometric measures for structures characterized by unitarily invariant norms, which is a large family covering most matrix norms of practical interest. Examples are provided to illustrate the utility of our theoretical development.