Sliding Window Persistence of Quasiperiodic Functions

TL;DR

This paper develops methods to analyze the persistent homology of sliding window point clouds derived from quasiperiodic functions, with applications in music dissonance detection.

Contribution

It introduces parameter optimization schemes and theoretical bounds for the persistent homology of quasiperiodic functions' sliding window sets.

Findings

Sliding window point clouds are dense in tori with dimension equal to the number of frequencies.

Theoretical lower bounds on Rips persistent homology are established.

Application demonstrated in music audio dissonance detection.

Abstract

A function is called quasiperiodic if its fundamental frequencies are linearly independent over the rationals. With appropriate parameters, the sliding window point clouds of such functions can be shown to be dense in tori with dimension equal to the number of independent frequencies. In this paper, we develop theoretical and computational techniques to study the persistent homology of such sets. Specifically, we provide parameter optimization schemes for sliding windows of quasiperiodic functions, and present theoretical lower bounds on their Rips persistent homology. The latter leverages a recent persistent K\"{u}nneth formula. The theory is illustrated via computational examples and an application to dissonance detection in music audio samples.