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Tropical Sufficient Statistics for Persistent Homology
Anthea Monod, Sara Kališnik Verovšek, Juan Ángel Patiño-Galindo and Lorin Crawford
In this paper, we show that an embedding in Euclidean space based on tropical algebraic geometry generates stable sufficient statistics for barcodes. Conventionally, barcodes are multiscale summaries of topological characteristics that capture the "shape" of data; however, in practice, they have complex structures which make them difficult to use in statistical settings. The sufficiency result presented in this work allows for classical probability distributions to be assumed on the tropicalized representation of barcodes. This makes a variety of parametric statistical inference methods amenable to barcodes, all while maintaining their initial interpretations. More specifically, we show that exponential family distributions may be assumed, and that likelihood functions for persistent homology may be constructed. We conceptually demonstrate sufficiency and illustrate its utility in persistent homology dimensions 0 and 1 with concrete parametric applications to HIV and avian influenza data.