Persistence Paths and Signature Features in Topological Data Analysis

Abstract

We introduce a new feature map for barcodes that arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in the tensor algebra of that vector space. The composition of these two operations -barcode to path, path to tensor series -results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness, and achieves state-of-the-art results on common classification benchmarks.Persistence paths and signature features. Notwithstanding their usefulness for certain tasks, barcodes are notoriously unsuitable for standard statistical inference because 1 arXiv:1806.00381v2 [stat.ML]

The Signature-Based Model for Early Detection of Sepsis from Electronic Health Records in the Intensive Care Unit

“…The signature S(X) completely characterises a path X up to tree-like equivalence and is invariant to reparameterisation [5]. The usefulness of signatures as features of sequential data was demonstrated theoretically for non-parametric hypothesis testing [4] and algebraic geometry [6] as well as in numerous machine learning applications [7], for example: in healthcare [8][9][10][11][12], finance [13], computer vision [14,15], topological data analysis [16] and deep signature learning [17].…”

A Survey of Topological Machine Learning Methods

“…Betti curves are advantageous because they permit the calculation of a mean curve, next to providing an easy-to-evaluate distance and kernel method. Chevyrev et al ( 2018 ) used this representation—and related “paths” derived from a persistence diagram and its representations—to solve classification tasks, using random forests and support vector machine classifiers. One drawback of the Betti curves is their limited expressive power.…”

A Persistent Homology Approach to Heart Rate Variability Analysis With an Application to Sleep-Wake Classification

“…For example, persistence landscapes (Bubenik, 2015) map persistence diagrams into a Banach space, specifically L p space. More examples include persistence image (Adams et al, 2017), generalized persistence landscapes (Berry et al, 2020), persistence path (Chevyrev et al, 2018), persistence codebook (Zelinski et al, 2020), persistence curves (Chung and Lawson, 2019), kernel based methods (Reininghaus et al, 2015;Kusano et al, 2016), and persistent entropy (Chintakunta et al, 2015;Atienza et al, 2019b). These methods have been studied and applied to different applications.…”