# Signal classification with a point process distance on the space of persistence diagrams

## Abstract

In this paper, we consider the problem of signal classification. First, the signal is translated into a persistence diagram through the use of delay-embedding and persistent homology. Endowing the data space of persistence diagrams with a metric from point processes, we show that it admits statistical structure in the form of Fréchet means and variances and a classification scheme is established. In contrast with the Wasserstein distance, this metric accounts for changes in small persistence and changes in cardinality. The classification results using this distance are benchmarked on both synthetic data and real acoustic signals and it is demonstrated that this classifier outperforms current signal classification techniques.

### Keywords

Classification of time series Data space of persistence diagrams Wasserstein metric Cardinality Persistent homology### Mathematics Subject Classification

62H30 62M10 54H99 62P30## Notes

### Acknowledgements

VM would like to thank the Army Research Office and its support via the Grant \(\#\) W911NF-17-1-0313 to VM. Both authors would like to thank Dr. Tung-Duong Tran-Luu for providing the Army Research Lab’s acoustic signal dataset and for useful discussions. The authors would also like to thank five anonymous reviewers for their comments, which substantially improved the manuscript.

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