Numerical CP Decomposition of Some Difficult Tensors

TL;DR

This paper introduces a numerical method based on constrained Levenberg-Marquardt optimization for CP decomposition of small tensors, particularly those representing small matrix multiplications, revealing their ranks and border ranks.

Contribution

It presents a novel numerical approach for CP decomposition of small tensors, including a new algorithm for 3x3 by 3x2 matrix multiplication with 15 multiplications.

Findings

Ranks of tensors for small matrix multiplications are identified.

A new algorithm for 3x3 and 3x2 matrix multiplication with 15 multiplications.

Numerical results confirm the effectiveness of the proposed method.

Abstract

In this paper, a numerical method is proposed for canonical polyadic (CP) decomposition of small size tensors. The focus is primarily on decomposition of tensors that correspond to small matrix multiplications. Here, rank of the tensors is equal to the smallest number of scalar multiplications that are necessary to accomplish the matrix multiplication. The proposed method is based on a constrained Levenberg-Marquardt optimization. Numerical results indicate the rank and border ranks of tensors that correspond to multiplication of matrices of the size 2x3 and 3x2, 3x3 and 3x2, 3x3 and 3x3, and 3x4 and 4x3. The ranks are 11, 15, 23 and 29, respectively. In particular, a novel algorithm for multiplying the matrices of the sizes 3x3 and 3x2 with 15 multiplications is presented.