Sketching Sparse Matrices, Covariances, and Graphs via Tensor Products
This paper considers the matter of recovering an unknown sparse $ptimes p$ matrix $X$ from an $mtimes m$ matrix $Y=AXB^T$ , where $A$ and $B$ are known $m times p$ matrices with $mll p$ . The most result shows that there exist constructions of the sketching matrices $A$ and $B$ so that even if $X$ has $mathcal O(p)$ nonzeros, it can be recovered precisely and efficiently employing a convex program so long as these nonzeros don't seem to be concentrated in any single row/column of $X$ . Furthermore, it suffices for the dimensions of $Y$ (the sketch dimension) to scale as $m = mathcal O(sqrt #text nonzeros in X times log p)$ . The results conjointly show that the recovery is sturdy and stable in the way that if $X$ is equal to a sparse matrix plus a perturbation, then the convex program we tend to propose produces an approximation w- th accuracy proportional to the dimensions of the perturbation. Unlike ancient results on sparse recovery, where the sensing matrix produces freelance measurements, our sensing operator is highly constrained (it assumes a tensor product structure). Therefore, proving recovery guarantees require nonstandard techniques. Indeed, our approach depends on a completely unique result concerning tensor products of bipartite graphs, which may be of freelance interest. This drawback is motivated by the subsequent application, among others. Consider a $p times n$ data matrix $D$ , consisting of $n$ observations of $p$ variables. Assume that the correlation matrix $X:=DD^T$ is (approximately) sparse in the way that every of the $p$ variables is considerably correlated with only a few others. Our results show that these important correlations can be detected whether or not we have access to solely a sketch of the info $S=AD$ with $A in R^mtimes p$ .
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