  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.
