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• W2964307931 abstract "Let $$mathcal {P}$$ be a set of $$n$$ random points in $$mathbb {R}^d$$ , generated from a probability measure on a $$m$$ -dimensional manifold $$mathcal {M}subset mathbb {R}^d$$ . In this paper we study the homology of $$mathcal {U}(mathcal {P},r)$$ —the union of $$d$$ -dimensional balls of radius $$r$$ around $$mathcal {P}$$ , as $$nrightarrow infty $$ , and $$rrightarrow 0$$ . In addition we study the critical points of $$d_{mathcal {P}}$$ —the distance function from the set $$mathcal {P}$$ . These two objects are known to be related via Morse theory. We present limit theorems for the Betti numbers of $$mathcal {U}(mathcal {P},r)$$ , as well as for number of critical points of index $$k$$ for $$d_{mathcal {P}}$$ . Depending on how fast $$r$$ decays to zero as $$n$$ grows, these two objects exhibit different types of limiting behavior. In one particular case ( $$n r^m ge C log n$$ ), we show that the Betti numbers of $$mathcal {U}(mathcal {P},r)$$ perfectly recover the Betti numbers of the original manifold $$mathcal {M}$$ , a result which is of significant interest in topological manifold learning." @default.
• W2964307931 created "2019-07-30" @default.
• W2964307931 creator A5070671339 @default.
• W2964307931 creator A5090478378 @default.
• W2964307931 date "2014-03-22" @default.
• W2964307931 modified "2023-09-26" @default.
• W2964307931 title "The topology of probability distributions on manifolds" @default.
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• W2964307931 doi "https://doi.org/10.1007/s00440-014-0556-x" @default.
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