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 be...
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Main Authors | , |
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Format | Journal Article |
Language | English |
Published |
15.12.2021
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Subjects | |
Online Access | Get full text |
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Summary: | 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
with CPD approximation and computation is 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. |
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DOI: | 10.48550/arxiv.2112.08283 |