Sketching for Kronecker Product Regression and P-splines

TensorSketch is an oblivious linear sketch introduced in (Pagh, 2013) and later used in (Pham and Pagh, 2013) in the context of SVMs for polynomial kernels. It was shown in (Avron et al., 2014) that TensorSketch provides a subspace embedding, and therefore can be used for canonical correlation analysis, low rank approximation, and principal component regression for the polynomial kernel. We take TensorSketch outside of the context of polynomials kernels, and show its utility in applications in which the underlying design matrix is a Kronecker product of smaller matrices. This allows us to solve Kronecker product regression and non-negative Kronecker product regression, as well as regularized spline regression. Our main technical result is then in extending TensorSketch to other norms. That is, TensorSketch only provides input sparsity time for Kronecker product regression with respect to the 2-norm. We show how to solve Kronecker product regression with respect to the 1-norm in time sublinear in the time required for computing the Kronecker product, as well as for more general p-norms.