Low-Rank Matrix Recovery from Row-and-Column Affine Measurements
This addresses a matrix recovery problem for applications where standard methods fail, offering a new measurement scheme and algorithm, but it appears incremental as it builds on existing SVD and least-squares techniques.
The paper tackles the problem of low-rank matrix recovery using row-and-column affine measurements, proposing an algorithm based on SVD and least-squares that achieves exact recovery with minimal measurements in noiseless cases and provides performance guarantees for noisy cases, with simulations showing improved speed and comparable or better accuracy.
We propose and study a row-and-column affine measurement scheme for low-rank matrix recovery. Each measurement is a linear combination of elements in one row or one column of a matrix $X$. This setting arises naturally in applications from different domains. However, current algorithms developed for standard matrix recovery problems do not perform well in our case, hence the need for developing new algorithms and theory for our problem. We propose a simple algorithm for the problem based on Singular Value Decomposition ($SVD$) and least-squares ($LS$), which we term \alg. We prove that (a simplified version of) our algorithm can recover $X$ exactly with the minimum possible number of measurements in the noiseless case. In the general noisy case, we prove performance guarantees on the reconstruction accuracy under the Frobenius norm. In simulations, our row-and-column design and \alg algorithm show improved speed, and comparable and in some cases better accuracy compared to standard measurements designs and algorithms. Our theoretical and experimental results suggest that the proposed row-and-column affine measurements scheme, together with our recovery algorithm, may provide a powerful framework for affine matrix reconstruction.