Tensor-structured sketching for constrained least squares

Abstract

In this work, we study a tensor-structured random sketching matrix to project a largescale convex optimization problem to a much lower-dimensional counterpart, which leads to huge memory and computation savings. We show that while maintaining the prediction error between a random estimator and the true solution with high probability, the dimension of the projected problem obtains optimal dependence in terms of the geometry of the constraint set. Moreover, the tensor structure and sparsity pattern of the structured random matrix yields extra computational advantage. Our analysis is based on probability chaining theory, which allows us to obtain an almost sharp estimate for the sketching dimension of convex optimization problems. Consequences of our main result are demonstrated in a few concrete examples, including unconstrained linear regressions and sparse recovery problems.

Johnson-Lindenstrauss Embeddings with Kronecker Structure

“…Kronecker products of independent Gaussian vectors, and proved embedding properties for such constructions for Kronecker products of order d = 2. The paper [2] extended the analysis beyond d = 2, and [15] further refined and extended these results in the context of sketching constrained least squares problems.…”