Jointly low-rank and bisparse recovery: Questions and partial answers

Analysis and Applications

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.