Fast and Guaranteed Tensor Decomposition via Sketching

TL;DR

This paper introduces fast, randomized tensor CP decomposition algorithms using sketching techniques, enabling efficient computation without explicitly forming large tensors, applicable to both sparse and dense data.

Contribution

The paper presents novel tensor-specific sketching methods, including FFT-based contractions and colliding hashes, to accelerate tensor decomposition algorithms.

Findings

Achieves faster tensor CP decomposition without sacrificing accuracy

Applicable to both sparse and dense tensors with consistent quality

Demonstrates competitive results in topic modeling applications

Abstract

Tensor CANDECOMP/PARAFAC (CP) decomposition has wide applications in statistical learning of latent variable models and in data mining. In this paper, we propose fast and randomized tensor CP decomposition algorithms based on sketching. We build on the idea of count sketches, but introduce many novel ideas which are unique to tensors. We develop novel methods for randomized computation of tensor contractions via FFTs, without explicitly forming the tensors. Such tensor contractions are encountered in decomposition methods such as tensor power iterations and alternating least squares. We also design novel colliding hashes for symmetric tensors to further save time in computing the sketches. We then combine these sketching ideas with existing whitening and tensor power iterative techniques to obtain the fastest algorithm on both sparse and dense tensors. The quality of approximation under our method does not depend on properties such as sparsity, uniformity of elements, etc. We apply the method for topic modeling and obtain competitive results.