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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)

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)

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)

The goal of this paper is to describe all closed, aspherical Riemannian manifolds M whose universal covers M have have a nontrivial amount of symmetry. By this we mean that Isom(M) is not discrete. By the well-known theorem of Myers-Steenrod [MS], this condition is equivalent to [Isom(M) : π 1 (M)] = ∞. Also note that if any cover of M has a nondiscrete… (More)

Let S = S g,n be a connected, orientable surface of genus g ≥ 0 with n ≥ 0 punctures. Let Teich(S) denote the corresponding Teichmüller space, and let Mod(S) denote the mapping class group of S. Understanding the analogy of Teich(S) with symmetric spaces is a well-known theme. Recall that a complete Riemannian manifold X is symmetric if it is symmetric at… (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)

In this paper, we study lower bounds on the K-theory of the maximal C *-algebra of a discrete group based on the amount of torsion it contains. We call this the finite part of the operator K-theory and give a lower bound that is valid for a large class of groups, called the " finitely embeddable groups ". The class of finitely embeddable groups includes all… (More)

It is well known that the signature operator on a manifold defines a K-homology class which is an orientation after inverting 2. Here we address the following puzzle: what is this class localized at 2, and what special properties does it have? Our answers include the following: • the K-homology class ∆ M of the signature operator is a bordism invariant; •… (More)