Implementation of linear maps with circulant matrices via modulo 2 rectifier circuits of bounded depth

TL;DR

This paper demonstrates how Boolean circulant matrices can be efficiently implemented using modulo 2 rectifier circuits with bounded depth, providing explicit complexity bounds for such implementations.

Contribution

It introduces a method to implement any Boolean circulant matrix with bounded-depth modulo 2 rectifier circuits and establishes complexity bounds depending on the circuit depth.

Findings

Implementation of circulant matrices with depth 2k-1 circuits

Implementation with depth 2k circuits and complexity O(n^{1+1/k} log^{-1/k} n

Explicit bounds for circuit complexity based on depth

Abstract

In the present note we show that for any positive integer k an arbitrary Boolean circulant matrix can be implemented via modulo 2 rectifier circuit of depth 2k-1 and complexity O(n^{1+1/k}), and also via circuit of depth 2k and complexity O(n^{1+1/k} log^{-1/k} n).