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

Aydin Buluç, Riley Murray, V. Bharadwaj, Laura Grigori, Osman Asif Malik, James Demmel

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.

Has companion code repository: https://github.com/vbharadwaj-bk/fast_tensor_leverage

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