Topological Consistency via Kernel Estimation

By Omer Bobrowski, Sayan Mukherjee, Jonathan E. Taylor. 2014

76 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/baf885557876f53f1f593591ff827b6627290505

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In Computational Theory and Mathematics

Citations in the corpus

On the topology of random complexes built over stationary point processes. (2012, unknown): “[9, 10, 14, 30, 39]).”
Topological Data Analysis (2016, background): “This approach has been suggested by Phillips et al. (2015); Chazal et al. (2014a); Bobrowski et al. (2014); Chung et al. (2009); Bubenik (2015).”
A roadmap for the computation of persistent homology (2015, methodology): “atistical analysis and machine-learning techniques. Such methods include persistence landscapes [24], using the space of algebraic functions [3], persistence images [39], and kernelization techniques [20,107,138,172]. See the papers [102,148] for applications of persistence landscapes. The package P”
An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists (2017, methodology): “For example, basic summaries are considered in Bonis et al. (2016) and combined with quantization and pooling methods to address non rigid shape analysis problems; Betti curves extracted from persistence diagrams are used with 1-dimensional Convolutional Neural Networks (CNN) to analyze time depende”
Maximally Persistent Cycles in Random Geometric Complexes (2015, background): “[11, 20, 21, 22, 23, 24, 32]) focuses on statistical inference using persistent homology, and include results about confidence intervals, consistency and robustness for topological estimation, subsampling and bootstrapping methods, and more.”