# The average condition number of most tensor rank decomposition problems is infinite

## TL;DR

This paper proves that the expected condition number for most tensor rank decompositions is infinite for higher ranks, highlighting the inherent computational complexity and implications for algorithm design.

## Contribution

It establishes that the average condition number is infinite for most tensor rank decompositions of rank 3 or higher, revealing fundamental complexity issues.

## Key findings

- Expected condition number is infinite for random rank-2 tensors under broad conditions.
- Expected angular condition number is finite for rank-2 tensors.
- Numerical experiments suggest higher ranks may also have finite angular condition numbers.

## Abstract

The tensor rank decomposition, or canonical polyadic decomposition, is the decomposition of a tensor into a sum of rank-1 tensors. The condition number of the tensor rank decomposition measures the sensitivity of the rank-1 summands with respect to structured perturbations. Those are perturbations preserving the rank of the tensor that is decomposed. On the other hand, the angular condition number measures the perturbations of the rank-1 summands up to scaling. We show for random rank-2 tensors that the expected value of the condition number is infinite for a wide range of choices of the density. Under a mild additional assumption, we show that the same is true for most higher ranks $r\geq 3$ as well. In fact, as the dimensions of the tensor tend to infinity, asymptotically all ranks are covered by our analysis. On the contrary, we show that rank-2 tensors have finite expected angular condition number. Based on numerical experiments, we conjecture that this could also be true for higher ranks. Our results underline the high computational complexity of computing tensor rank decompositions. We discuss consequences of our results for algorithm design and for testing algorithms computing tensor rank decompositions.