A Bilevel Optimization Method for Tensor Recovery Under Metric Learning Constraints

TL;DR

This paper introduces a bilevel optimization approach that jointly performs tensor completion and decomposition while learning a Mahalanobis distance metric, enhancing recovery accuracy by incorporating similarity information.

Contribution

It proposes a novel bilevel optimization framework connecting tensor recovery and metric learning, with a proven convergent algorithm for improved tensor completion and decomposition.

Findings

Significantly better performance on real data compared to previous methods.

Effectively incorporates similarity side information into tensor recovery.

Establishes a theoretical link between Mahalanobis metric and Tucker decomposition.

Abstract

Tensor completion and tensor decomposition are important problems in many domains. In this work, we leverage the connection between these problems to learn a distance metric that improves both decomposition and completion. We show that the optimal Mahalanobis distance metric for the completion task is closely related to the Tucker decomposition of the completed tensor. Then, we formulate a bilevel optimization problem to perform joint tensor completion and decomposition, subject to metric learning constraints. The metric learning constraints also allow us to flexibly incorporate similarity side information and coupled matrices, when available, into the tensor recovery process. We derive an algorithm to solve the bilevel optimization problem and prove its global convergence. When evaluated on real data, our approach performs significantly better compared to previous methods.