Crackle: The Homology of Noise

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 n of vertices grows, or can contain...

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Bibliographic Details
Published in:Discrete & computational geometry 2014-12, Vol.52 (4), p.680-704
Main Authors: Adler, Robert J., Bobrowski, Omer, Weinberger, Shmuel
Format: Article
Language:English
Subjects:
Analogies
Combinatorics
Computational Mathematics and Numerical Analysis
Gaussian
Geometry
Homology
Learning
Manifolds
Mathematics
Mathematics and Statistics
Noise
Random variables
Signal analysis
Temporal logic
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Summary: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 n 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 R d . 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.
ISSN:0179-5376
1432-0444
DOI:10.1007/s00454-014-9621-6