Sparse Approximation, List Decoding, and Uncertainty Principles

TL;DR

This paper explores list decoding in sparse approximation, providing the first combinatorial bounds on output list size and extending uncertainty principles to multiple solutions, inspired by coding theory.

Contribution

It introduces list versions of sparse approximation problems, establishing combinatorial bounds and generalizing existing threshold and uncertainty results.

Findings

First combinatorial bounds on list size in sparse approximation

Extension of uncertainty principles to list decoding scenarios

Lower bounds confirming the tightness of results

Abstract

We consider list versions of sparse approximation problems, where unlike the existing results in sparse approximation that consider situations with unique solutions, we are interested in multiple solutions. We introduce these problems and present the first combinatorial results on the output list size. These generalize and enhance some of the existing results on threshold phenomenon and uncertainty principles in sparse approximations. Our definitions and results are inspired by similar results in list decoding. We also present lower bound examples that bolster our results and show they are of the appropriate size.