Fast Low Rank column-wise Compressive Sensing.

We study the “Low Rank column-wise Compressive Sensing (LRcCS)” problem: recover an n × q rank-r matrix, ${X^ * } = \left[ {x_1^ *,x_2^ *, \ldots x_q^ * } \right]$, with r ≪ min(n,q), from ${y_k}: = {A_k}x_k^ *,k \in [q]$, when y k is an m-length vector with m < n. The matrices A k are known and mutually independent for different k. Even though many other LR recovery problems have been extensively studied, this problem has received little attention. We introduce a novel gradient descent (GD) based solution called altGDmin, and show that, if all entries of all A k s are i.i.d. Gaussian, and if the right singular vectors of X ^∗ satisfy the incoherence assumption, then ϵaccurate recovery of X ^∗ is possible with mq > C(n+q)r ² log(1/ϵ) total scalar samples and O(mqnrlog(1/ϵ)) time. Compared to existing work, to our best knowledge, this is the fastest solution and, for $\in < 1/\sqrt r$, it also has the best sample complexity.