Journal
DISCRETE & COMPUTATIONAL GEOMETRY
Volume 52, Issue 4, Pages 680-704Publisher
SPRINGER
DOI: 10.1007/s00454-014-9621-6
Keywords
Cech complex; Random complexes; Persistent homology; Random Betti numbers
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Funding
- AFOSR [FA8655-11-1-3039, FA9550-11-1-0216]
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We study the homology of simplicial complexes built via deterministic rules from a random set of vertices. In particular, we show that, depending on the randomness that generates the vertices, the homology of these complexes can either become trivial as the number of vertices grows, or can contain more and more complex structures. The different behaviours are consequences of different underlying distributions for the generation of vertices, and we consider three illustrative examples, when the vertices are sampled from Gaussian, exponential, and power-law distributions in . We also discuss consequences of our results for manifold learning with noisy data, describing the topological phenomena that arise in this scenario as crackle, in analogy to audio crackle in temporal signal analysis.
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