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The canonical polyadic decomposition (CPD) of a low rank tensor plays a majorrole in data analysis and signal processing by allowing for unique recovery ofunderlying factors. However, it is well known that the low rank CPDapproximation problem is ill-posed. That is, a tensor may fail to have a bestrank $R$ CPD approximation when $R>1$.  This article gives deterministic bounds for the existence of best low ranktensor approximations over $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$.More precisely, given a tensor $\mathcal{T} \in \mathbb{K}^{I \times I \timesI}$ of rank $R \leq I$, we compute the radius of a Frobenius norm ball centeredat $\mathcal{T}$ in which best $\mathbb{K}$-rank $R$ approximations areguaranteed to exist. In addition we show that every $\mathbb{K}$-rank $R$tensor inside of this ball has a unique canonical polyadic decomposition. Thisneighborhood may be interpreted as a neighborhood of "mathematical truth" inwith 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 thanborder rank is defective in the sense of algebraic and geometric multiplicitiesfor joint generalized eigenvalues. Bounds for existence of best low rankapproximations are then obtained by establishing perturbation theoretic resultsfor the joint generalized eigenvalue problem. In this way we establish aconnection between existence of best low rank approximations and the tensorspectral norm. In addition we solve a "tensor Procrustes problem" whichexamines orthogonal compressions for pairs of tensors.  The main results of the article are illustrated by a variety of numericalexperiments.

Guarantees for existence of a best canonical polyadic approximation of a noisy low-rank tensor