A Super-Resolution Framework for Tensor Decomposition

TL;DR

This paper introduces a super-resolution framework for overcomplete tensor decomposition, leveraging continuous $ ext{l}_1$ minimization to achieve unique recovery of tensor factors under certain incoherence conditions.

Contribution

It develops a novel super-resolution approach for tensor decomposition, establishing incoherence conditions that guarantee uniqueness and identifiability of tensor factors.

Findings

Incoherence conditions ensure unique tensor factor recovery.

Random tensor factors satisfy incoherence conditions with high probability.

The framework links tensor decomposition to super-resolution techniques.

Abstract

This work considers a super-resolution framework for overcomplete tensor decomposition. Specifically, we view tensor decomposition as a super-resolution problem of recovering a sum of Dirac measures on the sphere and solve it by minimizing a continuous analog of the $\ell_1$ norm on the space of measures. The optimal value of this optimization defines the tensor nuclear norm. Similar to the separation condition in the super-resolution problem, by explicitly constructing a dual certificate, we develop incoherence conditions of the tensor factors so that they form the unique optimal solution of the continuous analog of $\ell_1$ norm minimization. Remarkably, the derived incoherence conditions are satisfied with high probability by random tensor factors uniformly distributed on the sphere, implying global identifiability of random tensor factors.