Smaller Low-Depth Circuits for Kronecker Powers

TL;DR

This paper presents new, smaller depth-2 linear circuits for matrices that are Kronecker powers of fixed matrices, achieving improved size bounds for various specific matrices like Walsh-Hadamard and disjointness matrices.

Contribution

It introduces novel constructions of depth-2 circuits with reduced size for Kronecker power matrices, surpassing previous bounds and generalizing to certain larger depths.

Findings

Depth-2 circuit size for Kronecker powers improved to O(N^{1.5 - a_q}) for all q>1.

For q=2, circuit size improved to O(N^{1.446}), better than previous O(N^{1.493}).

For Walsh-Hadamard, circuit size improved to O(N^{1.443}).

Abstract

We give new, smaller constructions of constant-depth linear circuits for computing any matrix which is the Kronecker power of a fixed matrix. A standard argument (e.g., the mixed product property of Kronecker products, or a generalization of the Fast Walsh-Hadamard transform) shows that any such $N \times N$ matrix has a depth-2 circuit of size $O(N^{1.5})$. We improve on this for all such matrices, and especially for some such matrices of particular interest: - For any integer $q > 1$ and any matrix which is the Kronecker power of a fixed $q \times q$ matrix, we construct a depth-2 circuit of size $O(N^{1.5 - a_q})$, where $a_q > 0$ is a positive constant depending only on $q$. No bound beating size $O(N^{1.5})$ was previously known for any $q>2$. - For the case $q=2$, i.e., for any matrix which is the Kronecker power of a fixed $2 \times 2$ matrix, we construct a depth-2 circuit of size $O(N^{1.446})$, improving the prior best size $O(N^{1.493})$ [Alman, 2021]. - For the Walsh-Hadamard transform, we construct a depth-2 circuit of size $O(N^{1.443})$, improving the prior best size $O(N^{1.476})$ [Alman, 2021]. - For the disjointness matrix (the communication matrix of set disjointness, or equivalently, the matrix for the linear transform that evaluates a multilinear polynomial on all $0/1$ inputs), we construct a depth-2 circuit of size $O(N^{1.258})$, improving the prior best size $O(N^{1.272})$ [Jukna and Sergeev, 2013]. Our constructions also generalize to improving the standard construction for any depth $\leq O(\log N)$. Our main technical tool is an improved way to convert a nontrivial circuit for any matrix into a circuit for its Kronecker powers. Our new bounds provably could not be achieved using the approaches of prior work.