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We investigate the problem of recovering jointly $r$-rank and $s$-bisparsematrices from as few linear measurements as possible, considering arbitrarymeasurements 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, weinvestigate the possibility of achieving practical recovery via aniterative-hard-thresholding algorithm when $m \asymp r s^\gamma \ln(en/s)$ forsome exponent $\gamma > 0$. We show that this is feasible for $\gamma = 2$, andthat the proposed analysis cannot cover the case $\gamma \leq 1$. The precisevalue 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 jointlylow-rank and bisparse structure. Some related questions are partially answeredin passing. For rank-one measurements, we suggest on arcane grounds aniterative-hard-thresholding algorithm modified to exploit the nonstandardrestricted isometry property obeyed by this type of measurements.

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