Fast ultrametric matrix-vector multiplication

TL;DR

This paper introduces a novel tree-based encoding for ultrametric matrices that enables matrix-vector multiplication in linear time, significantly improving computational efficiency for large-scale problems.

Contribution

The authors develop a quadratic-time encoding method for ultrametric matrices into tree structures, facilitating linear-time matrix-vector multiplication.

Findings

Encoding ultrametric matrices as trees is efficient.

Matrix-vector multiplication is performed in linear time.

Empirical results show practical performance improvements.

Abstract

We study the properties of ultrametric matrices aiming to design methods for fast ultrametric matrix-vector multiplication. We show how to encode such a matrix as a tree structure in quadratic time and demonstrate how to use the resulting representation to perform matrix-vector multiplications in linear time. Accompanying this article, we provide an implementation of the proposed algorithms and present empirical results on their practical performance.