# Sketching Sparse Matrices, Covariances, and Graphs via Tensor Products

PROJECT TITLE:

Sketching Sparse Matrices, Covariances, and Graphs via Tensor Products

ABSTRACT:

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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