Estimation of low-rank tensors via convex optimization

TL;DR

This paper introduces three convex optimization methods for estimating low-rank tensor decompositions, automatically determining the rank and improving interpretability, with superior accuracy and efficiency demonstrated through experiments.

Contribution

It presents novel convex optimization approaches for tensor estimation that automatically infer rank and enhance interpretability, outperforming traditional methods.

Findings

More accurate predictive performance

Faster computation times

More reliable multilinear structure recovery

Abstract

In this paper, we propose three approaches for the estimation of the Tucker decomposition of multi-way arrays (tensors) from partial observations. All approaches are formulated as convex minimization problems. Therefore, the minimum is guaranteed to be unique. The proposed approaches can automatically estimate the number of factors (rank) through the optimization. Thus, there is no need to specify the rank beforehand. The key technique we employ is the trace norm regularization, which is a popular approach for the estimation of low-rank matrices. In addition, we propose a simple heuristic to improve the interpretability of the obtained factorization. The advantages and disadvantages of three proposed approaches are demonstrated through numerical experiments on both synthetic and real world datasets. We show that the proposed convex optimization based approaches are more accurate in predictive performance, faster, and more reliable in recovering a known multilinear structure than conventional approaches.