Refined methods for the identifiability of tensors

By Cristiano Bocci, Luca Chiantini, Giorgio Ottaviani. Annali di Matematica Pura ed Applicata, 2013

69 citations recorded by the source corpus.

We prove that the general tensor of size 2n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^n$$\end{document} and rank k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document} has a unique decomposition as the sum of decomposable tensors if k≤0.99972nn+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\le 0.9997\frac{2^n}{n+1}$$\end{document} (the constant 1 being the optimal value). Similarly, the general tensor of size 3n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3^n$$\end{document} and rank k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document} has a unique decomposition as the sum of decomposable tensors if k≤0.9983n2n+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\le 0.998\frac{3^n}{2n+1}$$\end{document} (the constant 1 being the optimal value). Some results of this flavor are obtained for tensors of any size, but the explicit bounds obtained are weaker.

Subfields: Artificial Intelligence, Computational Mathematics

View Paper: https://www.semanticscholar.org/paper/c44b078a7d9fd9c1097204c986e1626429d3ef5f

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Citations in the corpus

The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition (2018, background): “Generic identifiability is quite rare as a phenomenon, and it has been largely investigated; in particular, we refer to [200, 198, 199, 197, 201, 202, 189, 12].”
On generic identifiability of symmetric tensors of subgeneric rank (2015, background): “1 is based on the study of the geometric concepts of weak defectivity, developed in [8], and tangentially weak defectivity, developed in [6].”