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