Topological consistency via kernel estimation
arXiv:1407.5272 · doi:10.3150/15-BEJ744
Abstract
We introduce a consistent estimator for the homology (an algebraic structure representing connected components and cycles) of level sets of both density and regression functions. Our method is based on kernel estimation. We apply this procedure to two problems: (1) inferring the homology structure of manifolds from noisy observations, (2) inferring the persistent homology (a multi-scale extension of homology) of either density or regression functions. We prove consistency for both of these problems. In addition to the theoretical results, we demonstrate these methods on simulated data for binary regression and clustering applications.
Published at http://dx.doi.org/10.3150/15-BEJ744 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)
References in corpus (8)
- Consistency of spectral clustering
- Statistical topological data analysis using persistence landscapes
- Confidence sets for persistence diagrams
- Categorification of persistent homology
- Geometric Inference on Kernel Density Estimates
- Crackle: The Persistent Homology of Noise
- Robust statistics, hypothesis testing, and confidence intervals for persistent homology on metric measure spaces
- Tight Lower Bounds for Homology Inference
Cited by in corpus (16)
- A roadmap for the computation of persistent homology
- Robust Topological Inference: Distance To a Measure and Kernel Distance
- On the topology of random complexes built over stationary point processes
- Modeling and replicating statistical topology, and evidence for CMB non-homogeneity
- Probabilistic Convergence and Stability of Random Mapper Graphs
- Geometric Inference on Kernel Density Estimates
- DTM-based Filtrations
- Random Simplicial Complexes: Models and Phenomena
- A Topological Framework for Identifying Phenomenological Bifurcations in Stochastic Dynamical Systems
- Generalized Cluster Trees and Singular Measures
- Nonparametric Estimation of Probability Density Functions of Random Persistence Diagrams
- Nonparametric Estimation of Surface Integrals on Level Sets
- Topological data analysis approaches to uncovering the timing of ring structure onset in filamentous networks
- Box Filtration
- On the contractibility of random Vietoris-Rips complexes
- Nonparametric Confidence Regions for Level Sets: Statistical Properties and Geometry