The topology of probability distributions on manifolds

@article{Bobrowski2013TheTO,
  title={The topology of probability distributions on manifolds},
  author={Omer Bobrowski and Sayan Mukherjee},
  journal={Probability Theory and Related Fields},
  year={2013},
  volume={161},
  pages={651-686}
}
Let $$\mathcal {P}$$P be a set of $$n$$n random points in $$\mathbb {R}^d$$Rd, generated from a probability measure on a $$m$$m-dimensional manifold $$\mathcal {M}\subset \mathbb {R}^d$$M⊂Rd. In this paper we study the homology of $$\mathcal {U}(\mathcal {P},r)$$U(P,r)—the union of $$d$$d-dimensional balls of radius $$r$$r around $$\mathcal {P}$$P, as $$n\rightarrow \infty $$n→∞, and $$r\rightarrow 0$$r→0. In addition we study the critical points of $$d_{\mathcal {P}}$$dP—the distance function… 
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References

SHOWING 1-10 OF 67 REFERENCES
Collapsibility and Vanishing of Top Homology in Random Simplicial Complexes
TLDR
It is shown that there exists a constant $$\gamma _d< c_d c-d$$ and a fixed field $$\mathbb{F }$$, asymptotically almost surely $$H_d(Y;\mathBB{F }) \ne 0$$, and conjecture this bound to be sharp.
Distance Functions, Critical Points, and Topology for Some Random Complexes
For a finite set of points $P$ in $R^d$, the function $d_P:R^d \to R_+$ measures Euclidean distance to the set $P$. We study the number of critical points of $d_P$ when $P$ is random. In particular,
Homological Connectivity Of Random 2-Complexes
TLDR
It is shown that for any function ω(n) that tends to infinity, H_{1) is the first homology group of Y with mod 2 coefficients.
Topology of random clique complexes
Crackle: The Persistent 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
A Topological View of Unsupervised Learning from Noisy Data
TLDR
It is shown that if the variance of the Gaussian noise is small in a certain sense, then the homology can be learned with high confidence by an algorithm that has a weak (linear) dependence on the ambient dimension.
Complexity of random smooth functions of many variables
Can one count the number of critical points for random smooth functions of many variables? How complex is a typical random smooth function? How complex is the topology of its level sets? We study
Statistical topology via Morse theory, persistence and nonparametric estimation
In this paper we examine the use of topological methods for multivariate statistics. Using persistent homology from computational algebraic topology, a random sample is used to construct estimators
A statistical approach to persistent homology
TLDR
Using statistical estimators for samples from certain families of distributions, it is shown that the persistent homology of the underlying distribution can be recovered.
Random fields of multivariate test statistics, with applications to shape analysis
Our data are random fields of multivariate Gaussian observations, and we fit a multivariate linear model with common design matrix at each point. We are interested in detecting those points where
...
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