Convergence of the alternating least squares algorithm for CP tensor decompositions

TL;DR

This paper provides a detailed convergence analysis of the alternating least squares algorithm for CP tensor decompositions, establishing polynomial and linear convergence rates for specific tensor classes and proposing acceleration techniques.

Contribution

It offers the first explicit quantitative local convergence theorems for CP-AltLS on orthogonally and incoherently decomposable tensors, including constructive analysis and practical acceleration methods.

Findings

CP-AltLS converges polynomially for orthogonally decomposable tensors.

CP-AltLS converges linearly for incoherently decomposable tensors.

Numerical experiments confirm the theoretical convergence rates.

Abstract

The alternating least squares (ALS/AltLS) method is a widely used algorithm for computing the CP decomposition of a tensor. However, its convergence theory is still incompletely understood. In this paper, we prove explicit quantitative local convergence theorems for CP-AltLS applied to orthogonally decomposable and incoherently decomposable tensors. Specifically, we show that CP-AltLS converges polynomially with order $N-1$ for $N$th-order orthogonally decomposable tensors and linearly for incoherently decomposable tensors, with convergence being measured in terms of the angles between the factors of the exact tensor and those of the approximate tensor. Unlike existing results, our analysis is both quantitative and constructive, applying to standard CP-AltLS and accommodating factor matrices with small but nonzero mutual coherence, while remaining applicable to tensors of arbitrary rank. We also confirm these rates of convergence numerically and investigate accelerating the convergence of CP-AltLS using an SVD-based coherence reduction scheme.