Approximating Element-Wise Functions of Matrix with Improved Streaming Randomized SVD

Abstract

The element-wise functions of a matrix are widely used in machine learning. For the applications with large matrices, efficiently computing the matrix-vector multiplication of matrix element-wise function without explicitly constructed matrix is very desired. In this work, we aim to develop an efficient low-rank approximation of the element-wise function of matrix with the time/memory cost linear to the matrix dimension. We first propose a sparse-sign streaming randomized SVD (ssrSVD) algorithm based on a streaming singular value decomposition (SVD) algorithm and the sparse-sign random projection for the approximation of element-wise function of general asymmetric matrix. For symmetric positive semi-definite (SPSD) matrix, for which the existing Nyström [1] and FastSPSD [2] method do not perform well if the matrix's singular value decays slowly, we propose a theoretically proved shift skill to improve the approximation accuracy. Combining with the ssrSVD, we obtain the sparse-sign streaming SPSD matrix approximation with shift (S3SPSD) algorithm. Experiments are carried out to evaluate the proposed algorithms' performance in approximating element-wise functions of matrix. With the color transfer task based on the Sinkhorn algorithm, the ssrSVD algorithm largely reduces the approximation error (up to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$10^{5}\times$</tex> ) compared with the state-of-the-art baselines, and results in high-quality color transfer result. For the kernel matrix approximation, the proposed S3SPSD algorithm also consistently outperforms the state-of-the-art baselines. Experimental results finally validate the linear time complexity of the proposed algorithms.