An exact, cache-localized algorithm for the sub-quadratic convolution of hypercubes

TL;DR

This paper introduces an exact, cache-efficient algorithm for hypercube convolution that operates in sub-quadratic time, outperforming traditional FFT-based methods for high-dimensional data.

Contribution

The paper presents a novel sub-quadratic algorithm for exact hypercube convolution, improving efficiency over existing FFT-based approaches in high-dimensional settings.

Findings

Outperforms FFTPACK and FFTW in hypercube convolution tasks

Enables sub-quadratic algorithms for vector convolution variants

Efficiently handles high-dimensional hypercube data

Abstract

Fast multidimensional convolution can be performed naively in quadratic time and can often be performed more efficiently via the Fourier transform; however, when the dimensionality is large, these algorithms become more challenging. A method is proposed for performing exact hypercube convolution in sub-quadratic time. The method outperforms FFTPACK, called via numpy, and FFTW, called via pyfftw) for hypercube convolution. Embeddings in hypercubes can be paired with sub-quadratic hypercube convolution method to construct sub-quadratic algorithms for variants of vector convolution.