  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 the (possibly
high-rank) matrix [f(Aij)]. To this end, we propose an efficient
sketching-based algorithm whose complexity is significantly lower than the
number of entries of A, i.e., it runs without accessing all entries of [f(Aij)]
explicitly. The main idea underlying our method is to combine a polynomial
approximation of f with the existing tensor sketch scheme for approximating
monomials of entries of A. To balance the errors of the two approximation
components in an optimal manner, we propose a novel regression formula to find
polynomial coefficients given A and f. In particular, we utilize a
coreset-based regression with a rigorous approximation guarantee. Finally, we
demonstrate the applicability and superiority of the proposed scheme under
various machine learning tasks.
