# Jointly Low-Rank and Bisparse Recovery: Questions and Partial Answers

## TL;DR

This paper explores the theoretical limits and practical algorithms for recovering matrices that are both low-rank and bisparse from limited linear measurements, addressing open questions about measurement efficiency and algorithmic feasibility.

## Contribution

It provides bounds on the number of measurements needed for recovery and analyzes the potential of iterative-hard-thresholding algorithms under different measurement models.

## Key findings

- Theoretically, $m hickapprox r s \, \ln(en/s)$ measurements suffice for recovery.
- Practical iterative-hard-thresholding algorithms are feasible when $m \thickapprox r s^2 \ln(en/s)$ for certain conditions.
- Open questions remain about the optimal exponent $\gamma$ for measurement complexity and head projections.

## Abstract

We investigate the problem of recovering jointly $r$-rank and $s$-bisparse matrices from as few linear measurements as possible, considering arbitrary measurements as well as rank-one measurements. In both cases, we show that $m \asymp r s \ln(en/s)$ measurements make the recovery possible in theory, meaning via a nonpractical algorithm. In case of arbitrary measurements, we investigate the possibility of achieving practical recovery via an iterative-hard-thresholding algorithm when $m \asymp r s^\gamma \ln(en/s)$ for some exponent $\gamma > 0$. We show that this is feasible for $\gamma = 2$, and that the proposed analysis cannot cover the case $\gamma \leq 1$. The precise value of the optimal exponent $\gamma \in [1,2]$ is the object of a question, raised but unresolved in this paper, about head projections for the jointly low-rank and bisparse structure. Some related questions are partially answered in passing. For rank-one measurements, we suggest on arcane grounds an iterative-hard-thresholding algorithm modified to exploit the nonstandard restricted isometry property obeyed by this type of measurements.