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Recently there has been a lot of interest in geometrically motivated approaches to data analysis in high dimensional spaces. We consider the case where data is drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space. We show how to " learn " the homology of the submanifold with high confidence. We discuss… (More)

- J Bryant, S Ferry, W Mio, S Weinberger
- 1993

We construct examples of nonresolvable generalized n-manifolds, n ≥ 6, with arbitrary resolution obstruction, homotopy equivalent to any simply connected, closed n-manifold. We further investigate the structure of generalized manifolds and present a program for understanding their topology. By a generalized n-manifold we will mean a finite-dimensional… (More)

In this paper, we take a topological view of unsupervised learning. From this point of view, clustering may be interpreted as trying to find the number of connected components of an underlying geometrically structured probability distribution in a certain sense that we will make precise. We construct a geometrically structured probability distribution that… (More)

- J Bryant, S Ferry, W Mio, S Weinberger
- 2005

We prove that if X n , n ≥ 6, is a compact ANR homology n-manifold, we can blow up the singularities of X to obtain an ANR homology n-manifold with the disjoint disks property. More precisely, we show that there is an ANR homology n-manifold Y with the disjoint disks property and a cell-like map f : Y → X.

The main goal of this paper is to show that if a finite connected CW complex admits a continuous, symmetric, and unanimous choice function for some number n41 of agents, then the choice space is contractible. On the other hand, if one removes the finiteness, we give a complete characterization of the possible spaces; in particular, noncontractible spaces… (More)

We give some lower bounds on the description, sample, and computational complexities of the problems of computing dimension, homology, and topological type of a manifold, and detecting singularities for a polyhedron. The problem of making sense of large and high dimensional data sets is an extremely difficult and important one. In recent years, there has… (More)

- Samples, P Niyogi, S Smale, S Weinberger
- 2004

Recently there has been a lot of interest in geometrically motivated approaches to data analysis in high dimensional spaces. We consider the case where data is drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space. We show how to " learn " the homology of the submanifold with high confidence. We discuss… (More)

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