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Sliding window persistence of quasiperiodic functions

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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 Künneth formula. The theory is illustrated via computational examples and an application to dissonance detection in music audio samples.

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Notes

  1. Generously provided by Adam Huston.

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Acknowledgements

This work was partially supported by the National Science Foundation through grants DMS-1622301, CCF-2006661, and CAREER award DMS-1943758. The authors of this paper would like to thank Adam Huston for the audio recording of the brass horn. The first author would like to thank Rosemarie Bongers for discussions on some of the Harmonic Analysis aspects of this paper.

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  1. Department of Mathematics, University of Oklahoma, Norman, OK, USA

    Hitesh Gakhar

  2. Department of Mathematics and Khoury College of Computer Sciences, Northeastern University, Boston, MA, USA

    Jose A. Perea

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  1. Hitesh Gakhar
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Correspondence to Hitesh Gakhar.

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Gakhar, H., Perea, J.A. Sliding window persistence of quasiperiodic functions. J Appl. and Comput. Topology 8, 55–92 (2024). https://doi.org/10.1007/s41468-023-00136-7

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