On the computation of tensor functions under tensor-tensor multiplications with linear maps

TL;DR

This paper explores the computation of algebraic and non-algebraic tensor functions under tensor-tensor multiplication with linear maps, establishing complexity equivalences and proposing new tensor means and decompositions for applications like image compression.

Contribution

It introduces the asymptotic complexity equivalence of tensor and matrix multiplication, defines tensor geometric and Wasserstein means, and proposes a pseudo-SVD for linear maps, advancing tensor function computation methods.

Findings

Tensor polynomial evaluation complexity matches matrix multiplication complexity.

Tensor geometric mean can be computed via Riccati tensor equation.

Tensor geometric mean does not satisfy the resultantal identity.

Abstract

In this paper we study the computation of both algebraic and non-algebraic tensor functions under the tensor-tensor multiplication with linear maps. In the case of algebraic tensor functions, we prove that the asymptotic exponent of both the tensor-tensor multiplication and the tensor polynomial evaluation problem under this multiplication is the same as that of the matrix multiplication, unless the linear map is injective. As for non-algebraic functions, we define the tensor geometric mean and the tensor Wasserstein mean for pseudo-positive-definite tensors under the tensor-tensor multiplication with invertible linear maps, and we show that the tensor geometric mean can be calculated by solving a specific Riccati tensor equation. Furthermore, we show that the tensor geometric mean does not satisfy the resultantal (determinantal) identity in general, which the matrix geometric mean always satisfies. Then we define a pseudo-SVD for the injective linear map case and we apply it on image data compression.