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.

提出一种行列仿射测量方案以及相应的算法，可用于低秩矩阵恢复，具有较高的精度。

从行列仿射测量中恢复低秩矩阵