Polynomial Tensor Sketch for Element-wise Function of Low-Rank Matrix

This paper studies how to sketch element-wise functions of low-rank matrices. Formally, given low-rank matrix A = [Aij ] and scalar non-linear function f, we aim for finding an approximated low-rank representation of (high-rank) matrix [f(A_{ij})]. To this end, we propose an efficient sketch algorithm whose complexity is significantly lower than the number of entries of A, i.e., it runs without accessing all entries of [f(A_{ij})] explicitly. Our main idea is to combine a polynomial approximation on f with the existing tensor sketch scheme approximating monomials of entries of A. To balance errors of the two approximation components in an optimal manner, we address a novel regression formula to find polynomial coefficients given A and f. We demonstrate the applicability and superiority of the proposed scheme under the tasks of kernel SVM classification and optimal transport.