Stochastic Mirror Descent for Low-Rank Tensor Decomposition Under Non-Euclidean Losses

TL;DR

This paper introduces a scalable stochastic mirror descent algorithm for low-rank tensor decomposition under non-Euclidean loss functions, addressing computational challenges and demonstrating promising results in large-scale settings.

Contribution

It proposes a unified stochastic framework using tensor fiber sampling for non-Euclidean CPD, ensuring convergence and computational efficiency.

Findings

Achieves global convergence to stationary points.

Provides substantial computational savings.

Performs well on large-scale tensor data.

Abstract

This work considers low-rank canonical polyadic decomposition (CPD) under a class of non-Euclidean loss functions that frequently arise in statistical machine learning and signal processing. These loss functions are often used for certain types of tensor data, e.g., count and binary tensors, where the least squares loss is considered unnatural.Compared to the least squares loss, the non-Euclidean losses are generally more challenging to handle. Non-Euclidean CPD has attracted considerable interests and a number of prior works exist. However, pressing computational and theoretical challenges, such as scalability and convergence issues, still remain. This work offers a unified stochastic algorithmic framework for large-scale CPD decomposition under a variety of non-Euclidean loss functions. Our key contribution lies in a tensor fiber sampling strategy-based flexible stochastic mirror descent framework. Leveraging the sampling scheme and the multilinear algebraic structure of low-rank tensors, the proposed lightweight algorithm ensures global convergence to a stationary point under reasonable conditions. Numerical results show that our framework attains promising non-Euclidean CPD performance. The proposed framework also exhibits substantial computational savings compared to state-of-the-art methods.