Tensor decomposition beyond uniqueness, with an application to the minrank problem
Pascal Koiran, Rafael Oliveira
TL;DR
The paper investigates order-3 tensors $T \\in \\mathbb{K}^{m\\times n\\times p}$ in the undercomplete regime and generalizes Jennrich's uniqueness theorem by adopting a matrix-vector decomposition. It proves that under generic direct-sum conditions the minimum matrix-vector decomposition is unique and has rank equal to the tensor rank, accompanied by a randomized polynomial-time algorithm to compute it, and applies this to the minrank problem to recover all rank-minimizing matrices in certain generic subspaces. An application is shown for computing all matrices of minimum rank in subspaces where the hidden basis satisfies a direct-sum property, enabling efficient minrank computations over arbitrary fields. Conceptually, the work extends Jennrich-type decomposition methods to new regimes via generalized eigenvalue analysis of matrix pencils and randomized algorithms, broadening the practical reach of tensor decomposition in algebraic and computational settings.
Abstract
We prove a generalization to Jennrich's uniqueness theorem for tensor decompositions in the undercomplete setting. Our uniqueness theorem is based on an alternative definition of the standard tensor decomposition, which we call matrix-vector decomposition. Moreover, in the same settings in which our uniqueness theorem applies, we also design and analyze an efficient randomized algorithm to compute the unique minimum matrix-vector decomposition (and thus a tensor rank decomposition of minimum rank). As an application of our uniqueness theorem and our efficient algorithm, we show how to compute all matrices of minimum rank (up to scalar multiples) in certain generic vector spaces of matrices.