Counting tensor rank decompositions

TL;DR

This paper derives an explicit formula for the volume of approximate symmetric tensor rank decompositions within an error margin, extending to non-symmetric tensors and linking to matrix model partition functions.

Contribution

It provides a novel explicit formula for the volume of tensor decompositions averaged over unit norm tensors, including symmetric and non-symmetric cases, based on matrix model partition functions.

Findings

Explicit formula involving hypergeometric and power functions

Extension to non-symmetric tensor decompositions

Connection to matrix model partition functions

Abstract

The tensor rank decomposition is a useful tool for the geometric interpretation of the tensors in the canonical tensor model (CTM) of quantum gravity. In order to understand the stability of this interpretation, it is important to be able to estimate how many tensor rank decompositions can approximate a given tensor. More precisely, finding an approximate symmetric tensor rank decomposition of a symmetric tensor $Q$ with an error allowance $\Delta$ is to find vectors $\phi^i$ satisfying $\|Q-\sum_{i=1}^R \phi^i\otimes \phi^i\cdots \otimes \phi^i\|^2 \leq \Delta$. The volume of all possible such $\phi^i$ is an interesting quantity which measures the amount of possible decompositions for a tensor $Q$ within an allowance. While it would be difficult to evaluate this quantity for each $Q$, we find an explicit formula for a similar quantity by integrating over all $Q$ of unit norm. The expression as a function of $\Delta$ is given by the product of a hypergeometric function and a power function. We also extend the formula to generic decompositions of non-symmetric tensors. The derivation depends on the existence (convergence) of the partition function of a matrix model which appeared in the context of the CTM.