The arithmetic complexity of tensor contractions

TL;DR

This paper characterizes the algebraic complexity of tensor calculus, specifically tensor contractions, and links it to the complexity class VP, providing a new understanding of computational efficiency for tensor operations.

Contribution

It introduces a tensor-based generalization of matrix product formulas that precisely characterize the class VP, offering a natural complexity framework.

Findings

Tensor contractions capture VP complexity class

Provides a natural algebraic characterization of VP

Links tensor calculus to computational complexity theory

Abstract

We investigate the algebraic complexity of tensor calulus. We consider a generalization of iterated matrix product to tensors and show that the resulting formulas exactly capture VP, the class of polynomial families efficiently computable by arithmetic circuits. This gives a natural and robust characterization of this complexity class that despite its naturalness is not very well understood so far.