# Analysis of equivalence relation in joint sparse recovery

## TL;DR

This paper investigates the theoretical properties of l_p-minimization in joint sparse recovery, establishing conditions for exact recovery and analyzing the critical parameter p* that guarantees success.

## Contribution

It provides two main theoretical results: the existence of a critical p* for exact recovery in joint sparse problems and an analysis expression for p*.

## Key findings

- Existence of a constant p* ensuring recovery for 0<p<p*
- Analytical expression for the critical p*
- Validation through an example confirming theoretical results

## Abstract

The joint sparse recovery problem is a generalization of the single measurement vector problem which is widely studied in Compressed Sensing and it aims to recovery a set of jointly sparse vectors. i.e. have nonzero entries concentrated at common location. Meanwhile l_p-minimization subject to matrices is widely used in a large number of algorithms designed for this problem. Therefore the main contribution in this paper is two theoretical results about this technique. The first one is to prove that in every multiple systems of linear equation, there exists a constant p* such that the original unique sparse solution also can be recovered from a minimization in l_p quasi-norm subject to matrices whenever 0< p<p*. The other one is to show an analysis expression of such p*. Finally, we display the results of one example to confirm the validity of our conclusions.