The condition number of many tensor decompositions is invariant under Tucker compression

TL;DR

This paper proves that the condition number of various tensor decompositions remains unchanged under Tucker compression, enabling faster computation of sensitivity measures in practical tensor analysis.

Contribution

It establishes the invariance of the condition number under Tucker compression for multiple tensor decompositions, significantly improving computational efficiency.

Findings

Condition number invariance under Tucker compression

Speedup in computing condition numbers by four orders of magnitude

Application to large tensors in food science

Abstract

We characterise the sensitivity of several additive tensor decompositions with respect to perturbations of the original tensor. These decompositions include canonical polyadic decompositions, block term decompositions, and sums of tree tensor networks. Our main result shows that the condition number of all these decompositions is invariant under Tucker compression. This result can dramatically speed up the computation of the condition number in practical applications. We give the example of an $265\times 371\times 7$ tensor of rank $3$ from a food science application whose condition number was computed in $6.9$ milliseconds by exploiting our new theorem, representing a speedup of four orders of magnitude over the previous state of the art.