Optimal rates of convergence for persistence diagrams in Topological Data Analysis
Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field. In this paper, we study topological persistence in general metric spaces, with a statistical approach. We show that the use of persistent homology can be naturally considered in general statistical frameworks and persistence diagrams can be used as statistics with interesting convergence properties. Some numerical experiments are performed in various contexts to illustrate our results.
Code (0)
등록된 구현이 없습니다.
Tasks
Topological Data AnalysisSimilar Papers 제목 키워드 기반
$k$-Means Clustering for Persistent Homology
Persistent homology is a methodology central to topological data analysis that extracts and summarizes the topological features within a dataset as a persistence diagram; it has recently gained much popularity from its m…
ClusteringTopological Data AnalysisDifferentially Private Topological Data Analysis
This paper is the first to attempt differentially private (DP) topological data analysis (TDA), producing near-optimal private persistence diagrams. We analyze the sensitivity of persistence diagrams in terms of the bott…
SensitivityTopological Data AnalysisA Stable Cardinality Distance for Topological Classification
This work incorporates topological features via persistence diagrams to classify point cloud data arising from materials science. Persistence diagrams are multisets summarizing the connectedness and holes of given data. …
ClassificationGeneral ClassificationFuzzy c-Means Clustering for Persistence Diagrams
Persistence diagrams concisely represent the topology of a point cloud whilst having strong theoretical guarantees, but the question of how to best integrate this information into machine learning workflows remains open.…
BIG-bench Machine LearningClusteringModel SelectionTopological Data AnalysisRobust Persistence Diagrams using Reproducing Kernels
Persistent homology has become an important tool for extracting geometric and topological features from data, whose multi-scale features are summarized in a persistence diagram. From a statistical perspective, however, p…