Analysis and Approximation of the Canonical Polyadic Tensor Decomposition

TL;DR

This paper analyzes the least-squares functional of CP tensor decomposition, introduces a new projection-based approach, and proposes the centroid projection algorithm to improve initialization and speed up tensor decomposition methods.

Contribution

It introduces a reduced functional reformulation, new conditions for minimizer existence, and the centroid projection algorithm for efficient tensor decomposition initialization.

Findings

The centroid projection algorithm provides faster convergence.

Suboptimal solutions improve iterative CP algorithms.

The approach offers new theoretical insights into the LS functional.

Abstract

We study the least-squares (LS) functional of the canonical polyadic (CP) tensor decomposition. Our approach is based on the elimination of one factor matrix which results in a reduced functional. The reduced functional is reformulated into a projection framework and into a Rayleigh quotient. An analysis of this functional leads to several conclusions: new sufficient conditions for the existence of minimizers of the LS functional, the existence of a critical point in the rank-one case, a heuristic explanation of "swamping" and computable bounds on the minimal value of the LS functional. The latter result leads to a simple algorithm -- the Centroid Projection algorithm -- to compute suboptimal solutions of tensor decompositions. These suboptimal solutions are applied to iterative CP algorithms as initial guesses, yielding a method called centroid projection for canonical polyadic (CPCP) decomposition which provides a significant speedup in our numerical experiments compared to the standard methods.