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

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 ≍ rs 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 ≍ rs γ ln(en/s) for some exponent γ > 0. We show that this is feasible for γ = 2, and that the proposed analysis cannot cover the case γ ≤ 1. The precise value of the optimal exponent γ ∈ [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.

One-Bit Sensing of Low-Rank and Bisparse Matrices

“…Let us now establish the upper bound in (2). We first show that, with high probability on a Gaussian random map A, all pairs of matrices in a set K ⊆ B n×n F that are consistent under sgn A must be at most ε apart provided m is large compared to ε and to the intrinsic set complexity.…”

“…With Corollary 1 at hand, we can justify the upper bound in (2). For this purpose, consider ε > 0 making (9) an equality.…”

“…[r] (s) , but we stress from the onset that computing it is not a realistic task, see [2] for a related NP-hardness statement. We note also that X ′ need not be consistent, in the sense that sgn A(X ′ ) is not necessarily equal to y, but that it has the same structure as X.…”

“…While the projected back projection scheme (10) is impractical, changing the sensing map A : R n×n → R m allows for the design of a practical algorithm to recover matrices in Σ [r] (s) with provable recovery error decaying as a root of rs ln(n/s)/m. The strategy follows a multistep argument put forward in [12] for sparse phase retrieval and also imitated in [2] for low-rank and bisparse recovery from standard, i.e., not quantized, measurements. It consists in defining a sensing map A : R n×n → R m associated with matrices…”

“…Determining the minimal number of linear measurements allowing for uniform stable recovery of matrices in Σ [r] (s) is a difficult problem, e.g. [1] contains negative results for optimization-based recovery, while [2] showed, with some caveats, that Θ(rs ln(n/s)) measurement suffice. Here, we consider a similar problem for quantized linear measurements, hoping for more favorable conclusions in this case.…”