A concise proof of Kruskal's theorem on tensor decomposition

TL;DR

This paper presents a shorter, more concise proof of Kruskal's 1977 theorem on the essential uniqueness of tensor decompositions, which is fundamental in tensor analysis and statistical modeling.

Contribution

It provides a significantly shorter proof of Kruskal's theorem, improving upon previous proofs in clarity and brevity.

Findings

The new proof simplifies understanding of tensor decomposition uniqueness.

The proof confirms the conditions under which tensor decompositions are essentially unique.

It enhances the theoretical foundation for applications in statistical models and data analysis.

Abstract

A theorem of J. Kruskal from 1977, motivated by a latent-class statistical model, established that under certain explicit conditions the expression of a 3-dimensional tensor as the sum of rank-1 tensors is essentially unique. We give a new proof of this fundamental result, which is substantially shorter than both the original one and recent versions along the original lines.