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

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 K=R or K=C. More precisely, given a tensor T∈KI×I×I of rank R≤I, we compute the radius of a Frobenius norm ball centered at T in which best K-rank R approximations are guaranteed to exist. In addition we show that every 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.

Ketersediaan

Informasi Detail

Judul Seri

SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS; Vol.43 No.1 January-March 2022

No. Panggil

-

Penerbit

: .,

Deskripsi Fisik

p. 328-369

Bahasa

English

ISBN/ISSN

-

Klasifikasi

NONE

Tipe Isi

-

Tipe Media

-

Tipe Pembawa

-

Edisi

-

Subjek

EXISTENCE
LOW-RANK APPROXIMATION
CANONICAL POLYADIC DECOMPOSITION
MULTILINEAR ALGEBRA
BORDER RANK
TENSORS
GENERALIZED EIGENVALUE
BAUER--FIKE MATCHING DISTANCE

Info Detail Spesifik

https://doi.org/10.1137/20M1381046

Pernyataan Tanggungjawab

Eric Evert, Lieven De Lathauwer

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