A condition number for the tensor rank decomposition

TL;DR

This paper introduces a condition number for tensor rank decomposition that measures the sensitivity of the parameters to small changes in the tensor, linking it to the inverse of a singular value of Terracini's matrix.

Contribution

It defines and analyzes a new condition number for tensor rank decomposition, connecting it to the singular values of Terracini's matrix and exploring its properties.

Findings

The condition number equals the inverse of the least singular value of Terracini's matrix.

Basic properties of the condition number are investigated.

The condition number provides insights into the stability of tensor rank decomposition.

Abstract

The tensor rank decomposition problem consists of recovering the unique set of parameters representing a robustly identifiable low-rank tensor when the coordinate representation of the tensor is presented as input. A condition number for this problem measuring the sensitivity of the parameters to an infinitesimal change to the tensor is introduced and analyzed. It is demonstrated that the absolute condition number coincides with the inverse of the least singular value of Terracini's matrix. Several basic properties of this condition number are investigated.