Fast Exact Leverage Score Sampling from Khatri-Rao Products with Applications to Tensor Decomposition

Vivek Bharadwaj, Osman Asif Malik, Riley Murray, Laura Grigori, Aydin Buluc, James Demmel

Keywords: Tensor Decomposition, Leverage Scores, Randomized Linear Algebra, Sketching, Khatri-Rao Product, Sparse Tensors

TL;DR: We built a fast data structure to sample rows from a Khatri-Rao product according to its exact leverage scores, which leads to asymptotically faster and practically more accurate randomized CP decomposition.

Abstract: We present a data structure to randomly sample rows from the Khatri-Rao product of several matrices according to the exact distribution of its leverage scores. Our proposed sampler draws each row in time logarithmic in the height of the Khatri-Rao product and quadratic in its column count, with persistent space overhead at most the size of the input matrices. As a result, it tractably draws samples even when the matrices forming the Khatri-Rao product have tens of millions of rows each. When used to sketch the linear least-squares problems arising in Candecomp / PARAFAC decomposition, our method achieves lower asymptotic complexity per solve than recent state-of-the-art methods. Experiments on billion-scale sparse tensors and synthetic data validate our theoretical claims, with our algorithm achieving higher accuracy than competing methods as the decomposition rank grows.

Submission Number: 1958