Algebraic compressed sensing

TL;DR

This paper introduces algebraic compressed sensing, using algebraic geometry to analyze the conditions for signal recovery, including existence and uniqueness, with results applicable to low-rank matrix and tensor problems.

Contribution

It develops a general algebraic framework for compressed sensing problems, providing optimal bounds on measurements needed for recovery based on model dimension.

Findings

Most bounds on measurements are optimal.

Results apply to low-rank matrices and tensors.

Provides a comprehensive algebraic analysis of compressed sensing.

Abstract

We introduce the broad subclass of algebraic compressed sensing problems, where structured signals are modeled either explicitly or implicitly via polynomials. This includes, for instance, low-rank matrix and tensor recovery. We employ powerful techniques from algebraic geometry to study well-posedness of sufficiently general compressed sensing problems, including existence, local recoverability, global uniqueness, and local smoothness. Our main results are summarized in thirteen questions and answers in algebraic compressed sensing. Most of our answers concerning the minimum number of required measurements for existence, recoverability, and uniqueness of algebraic compressed sensing problems are optimal and depend only on the dimension of the model.