Efficient Matrix Sensing Using Rank-1 Gaussian Measurements

In this paper, we study the problem of low-rank matrix sensing where the goal is to reconstruct a matrix {\em exactly} using a small number of linear measurements. Existing methods for the problem either rely on measurement operators such as random element-wise sampling which cannot recover arbitrary low-rank matrices or require the measurement operator to satisfy the Restricted Isometry Property (RIP). However, RIP based linear operators are generally full rank and require large computation/storage cost for both measurement (encoding) as well as reconstruction (decoding).

In this paper, we propose simple rank-one Gaussian measurement operators for matrix sensing that are significantly less expensive in terms of memory and computation for both encoding and decoding. Moreover, we show that the matrix can be reconstructed exactly using a simple alternating minimization method as well as a nuclear-norm minimization method. Finally, we demonstrate the effectiveness of the measurement scheme vis-a-vis existing RIP based methods.