Uniqueness of Tensor Decompositions with Applications to Polynomial Identifiability

TL;DR

This paper presents a robust version of Kruskal's tensor decomposition theorem, enabling approximate recovery of tensor components under small errors and establishing polynomial identifiability for latent variable models.

Contribution

It introduces a robust form of Kruskal's theorem that allows for approximate tensor decomposition and identifiability under milder conditions than previously possible.

Findings

Robust tensor decomposition is achievable with small error bounds.

Polynomial sample complexity for parameter identifiability in latent models.

Extension of tensor methods beyond non-degeneracy conditions.

Abstract

We give a robust version of the celebrated result of Kruskal on the uniqueness of tensor decompositions: we prove that given a tensor whose decomposition satisfies a robust form of Kruskal's rank condition, it is possible to approximately recover the decomposition if the tensor is known up to a sufficiently small (inverse polynomial) error. Kruskal's theorem has found many applications in proving the identifiability of parameters for various latent variable models and mixture models such as Hidden Markov models, topic models etc. Our robust version immediately implies identifiability using only polynomially many samples in many of these settings. This polynomial identifiability is an essential first step towards efficient learning algorithms for these models. Recently, algorithms based on tensor decompositions have been used to estimate the parameters of various hidden variable models efficiently in special cases as long as they satisfy certain "non-degeneracy" properties. Our methods give a way to go beyond this non-degeneracy barrier, and establish polynomial identifiability of the parameters under much milder conditions. Given the importance of Kruskal's theorem in the tensor literature, we expect that this robust version will have several applications beyond the settings we explore in this work.