Notes on the complexity of coverings for Kronecker powers of symmetric matrices

TL;DR

This paper improves the understanding of coverings for Kronecker powers of symmetric matrices, providing a stronger proof and better bounds on the complexity of specific boolean matrices, advancing matrix complexity theory.

Contribution

It offers an alternative, stronger proof for symmetric matrices and improves the upper bound on the additive complexity of Kneser-Sierpinski matrices.

Findings

Improved upper bound on depth-2 additive complexity to O(N^{1.251})

Stronger proof for symmetric matrices' coverings

Enhanced understanding of matrix coverings for Kronecker powers

Abstract

In the present note, we study a new method of constructing efficient coverings for Kronecker powers of matrices, recently proposed by J. Alman, Y. Guan, A. Padaki [arXiv, 2022]. We provide an alternative proof for the case of symmetric matrices in a stronger form. As a consequence, the previously known upper bound on the depth-2 additive complexity of the boolean $N\times N$ Kneser-Sierpinski matrices is improved to $O(N^{1.251})$.