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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)

- ALEXANDER NABUTOVSKY, SHMUEL WEINBERGER
- 2000

This paper is devoted to large scale aspects of the geometry of the space of isometry classes of Riemannian metrics, with a 2-sided curvature bound, on a fixed compact smooth manifold of dimension at least five. Using a mix of tools from logic/computer science, and differential geometry and topology, we study the diameter functional and its critical points,… (More)

We develop a theory of tubular neighborhoods for the lower strata in manifold stratified spaces with two strata. In these topologically stratified spaces, manifold approximate fibrations and teardrops play the role that fibre bundles and mapping cylinders play in smoothly stratified spaces. Applications include the classification of neighborhood germs, the… (More)

We prove that, if M is a compact oriented manifold of dimension 4k + 3, where k > 0, such that π 1 (M) is not torsion-free, then there are infinitely many manifolds that are homotopic equivalent to M but not homeomorphic to it. To show the infinite size of the structure set of M , we construct a secondary invariant τ (2) : S(M) → R that coincides with the… (More)

Define the length of a finite presentation of a group G as the sum of lengths of all relators plus the number of generators. How large can be the kth Betti number b k (G) = rank H k (G) providing that G has length ≤ N and b k (G) is finite? We prove that for every k ≥ 3 the maximum b k (N) of kth Betti numbers of all such groups is an extremely rapidly… (More)

- Erik Guentner, Nigel Higson, Shmuel Weinberger
- 2003

Let K be a field. We show that every countable subgroup of GL(n, K) is uniformly embeddable in a Hilbert space. This implies that Novikov's higher signature conjecture holds for these groups. We also show that every countable subgroup of GL(2, K) admits a proper, affine isometric action on a Hilbert space. This implies that the Baum-Connes conjecture holds… (More)

- Robert J. Adler, Omer Bobrowski, Mathew S. Borman, Eliran Subag, Shmuel Weinberger
- 2010

We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more about manifold learning from random point cloud data, the algebraic structure of simplicial complexes determined by random vertices, and, in most detail, the… (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)