Fast Tensor Completion via Approximate Richardson Iteration

TL;DR

This paper introduces a novel lifting method for tensor completion that leverages structured tensor decomposition algorithms as blackbox subroutines, enabling faster, sublinear-time solutions with proven convergence.

Contribution

It presents a new lifting approach for tensor completion that allows the use of existing tensor decomposition algorithms to achieve faster, sublinear-time completion.

Findings

Empirical results show up to 100x speedup over direct methods.

The proposed algorithm has a proven convergence rate.

Applicable to real-world tensor completion tasks.

Abstract

We study tensor completion (TC) through the lens of low-rank tensor decomposition (TD). Many TD algorithms use fast alternating minimization methods to solve highly structured linear regression problems at each step (e.g., for CP, Tucker, and tensor-train decompositions). However, such algebraic structure is often lost in TC regression problems, making direct extensions unclear. This work proposes a novel lifting method for approximately solving TC regression problems using structured TD regression algorithms as blackbox subroutines, enabling sublinear-time methods. We analyze the convergence rate of our approximate Richardson iteration-based algorithm, and our empirical study shows that it can be 100x faster than direct methods for CP completion on real-world tensors.