Jointly Low-Rank and Bisparse Recovery : Questions and Partial Answers Article - 2020

Simon Foucart, Rémi Gribonval, Laurent Jacques, Holger Rauhut

Simon Foucart, Rémi Gribonval, Laurent Jacques, Holger Rauhut, « Jointly Low-Rank and Bisparse Recovery : Questions and Partial Answers  », Analysis and Applications, 2020, pp. 25—48. ISSN 0219-5305

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 the 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.

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