Counting tensor rank decompositions
Dennis Obster, Naoki Sasakura

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 with an error allowance is to find vectors satisfying . The volume of all possible such is an interesting quantity which measures the amount of possible decompositions for a tensor within an allowance. While it would be difficult to evaluate this quantity for each , we find an explicit formula for a similar quantity by integrating over all of unit norm. The…
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Taxonomy
TopicsTensor decomposition and applications · Black Holes and Theoretical Physics · Noncommutative and Quantum Gravity Theories
