Guarantees for Existence of a Best Canonical Polyadic Approximation of a Noisy Low-Rank Tensor

Abstract

The canonical polyadic decomposition (CPD) of a low-rank tensor plays a major role in data analysis and signal processing by allowing for unique recovery of underlying factors. However, it is well known that the low-rank CPD approximation problem is ill-posed. That is, a tensor may fail to have a best rank $R$ CPD approximation when $R>1$. This article gives deterministic bounds for the existence of best low-rank tensor approximations over ${\mathbb{K}}={\mathbb{R}}$ or ${\mathbb{K}}={\mathbb{C}}$. More precisely, given a tensor ${\mathcal T} \in {\mathbb{K}}^{I \times I \times I}$ of rank $R \leq I$, we compute the radius of a Frobenius norm ball centered at ${\mathcal T}$ in which best ${\mathbb{K}}$-rank $R$ approximations are guaranteed to exist. In addition we show that every ${\mathbb{K}}$-rank $R$ tensor inside of this ball has a unique canonical polyadic decomposition. This neighborhood may be interpreted as a neighborhood of “mathematical truth" in which CPD approximation and computation are well-posed. In pursuit of these bounds, we describe low-rank tensor decomposition as a “joint generalized eigenvalue" problem. Using this framework, we show that, under mild assumptions, a low-rank tensor which has rank strictly greater than border rank is defective in the sense of algebraic and geometric multiplicities for joint generalized eigenvalues. Bounds for existence of best low-rank approximations are then obtained by establishing perturbation theoretic results for the joint generalized eigenvalue problem. In this way we establish a connection between existence of best low-rank approximations and the tensor spectral norm. In addition we solve a “tensor Procrustes problem" which examines orthogonal compressions for pairs of tensors. The main results of the article are illustrated by a variety of numerical experiments.