Abstract
Topological data analysis (TDA) is a rising field in the intersection of mathematics,
statistics, and computer science/data science. The cornerstone of TDA is persistent
homology, which produces a summary of topological information called a persistence
diagram. To utilize machine and deep learning methods, these diagrams are summa
rized by transforming them into functions. In this paper we investigate the stability
and injectivity of a class of smooth, one-dimensional functional summaries called
Gaussian persistence curves.
introduction
ne of the main tools of topological data analysis (TDA) is persistent homology,
which measures how certain topological features of a data set appear and disappear
at different scales. This information can be stored and visualized in a concise format
called a persistence diagram.
Functional summaries play an important role in topological data analysis, as they
allow one to apply machine anddeeplearning techniques to analyze topological infor
mation contained in persistence diagrams. In [7] a new class of one-dimensional
smooth functional summaries was introduced called Gaussian persistence curves
(GPC’s). These functional summaries were built by combining (a slight variation of)
the persistence curve framework from [6] with the persistence surfaces construction
from [1]. The GPC construction produces “smoothed” versions of many well known
functional summaries, such as the betti curve, the life curve, and the midlife curve [7].
The “smoothed” curves are then less sensitive to small fluctuations in the diagrams.
Statistical properties of GPC’s and applications of GPC’s to the texture classification
of grey-scale images were studied in [7].
In this paper, we investigate the stability of GPC’s and the injectivity of both per
sistence surfaces and GPC’s. Loosely speaking, stability refers to the property that
small changes in diagrams correspond to small changes in the resulting summaries
andtheinjectivity ofasummaryimpliesthatthesummarycandistinguishbetweendis
tinct diagrams. We show that unweighted GPC’s are stable (Theorem 3.2) and almost
injective (Theorem 4.4) and that, under mild conditions, weighted GPC’s are stable
(Corollary 3.4 and Theorem 3.7). Furthermore we showthat the identifiability of finite
Gaussian mixture models implies that unweighted (and some weighted) persistence
surfaces are injective (Theorem 4.2).
Other summaries in topological data analysis include persistence landscapes [4],
the persistent entropy summary function [2], persistence silhouettes [5], persistence
surfaces and persistence images [1]. We refer to [3] for a review of the properties and
applications of these summaries. All of these other summaries are known to be stable,
but among them only the persistence landscapes are known to be injective. Note that
persistence landscapes can be viewed as a sequences of one-dimensional functions
and for any n ≥ 1 injectivity will fail if only the first n terms of the sequences are
considered.
The outline of this paper is as follows. In Section 2 we introduce Gaussian per
sistence curves and derive some basic properties and useful formulas. In Section 3
weapply these formulas to prove stability of unweighted GPC’s and certain weighted
GPC’s. Finally, in Section 4 we investigate the extent to which the functional sum
maries produced by persistence surfaces and GPC’s are injective.