Provable Tensor Factorization with Missing Data

TL;DR

This paper introduces a new alternating minimization method for low-rank tensor factorization with missing data, providing theoretical guarantees for exact recovery from a specific number of samples.

Contribution

It presents a novel algorithm with provable guarantees for tensor completion, extending spectral analysis and convergence proofs to tensors with missing entries.

Findings

Exact tensor recovery from O(n^{3/2} r^5 log^4 n) samples

Generalization of Szemerédi's result to tensor spectral analysis

Global convergence of the proposed alternating minimization method

Abstract

We study the problem of low-rank tensor factorization in the presence of missing data. We ask the following question: how many sampled entries do we need, to efficiently and exactly reconstruct a tensor with a low-rank orthogonal decomposition? We propose a novel alternating minimization based method which iteratively refines estimates of the singular vectors. We show that under certain standard assumptions, our method can recover a three-mode $n\times n\times n$ dimensional rank-$r$ tensor exactly from $O(n^{3/2} r^5 \log^4 n)$ randomly sampled entries. In the process of proving this result, we solve two challenging sub-problems for tensors with missing data. First, in the process of analyzing the initialization step, we prove a generalization of a celebrated result by Szemer\'edie et al. on the spectrum of random graphs. Next, we prove global convergence of alternating minimization with a good initialization. Simulations suggest that the dependence of the sample size on dimensionality $n$ is indeed tight.