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42
Geometric inference for probability measures
 FOCM
"... Data often comes in the form of a point cloud sampled from an unknown compact subset of Euclidean space. The general goal of geometric inference is then to recover geometric and topological features (eg. Betti numbers, normals) of this subset from the approximating point cloud data. In recent years, ..."
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Cited by 31 (7 self)
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Data often comes in the form of a point cloud sampled from an unknown compact subset of Euclidean space. The general goal of geometric inference is then to recover geometric and topological features (eg. Betti numbers, normals) of this subset from the approximating point cloud data. In recent years, it appeared that the study of distance functions allows to address many of these questions successfully. However, one of the main limitations of this framework is that it does not cope well with outliers nor with background noise. In this paper, we show how to extend the framework of distance functions to overcome this problem. Replacing compact subsets by measures, we introduce a notion of distance function to a probability distribution in Rd. These functions share many properties with classical distance functions, which make them suitable for inference purposes. In particular, by considering appropriate level sets of these distance functions, we show that it is possible to reconstruct offsets of sampled shapes with topological guarantees even in the presence of outliers. Moreover, in settings where empirical measures are considered these functions can be easily evaluated, making them of particular practical interest. 1
Perelman’s stability theorem
, 2007
"... Abstract. We give a proof of the celebrated stability theorem of Perelman stating that for a noncollapsing sequence Xi of Alexandrov spaces with curv � k GromovHausdorff converging to a compact Alexandrov space X, Xi is homeomorphic to X for all large i. 1. ..."
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Cited by 21 (0 self)
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Abstract. We give a proof of the celebrated stability theorem of Perelman stating that for a noncollapsing sequence Xi of Alexandrov spaces with curv � k GromovHausdorff converging to a compact Alexandrov space X, Xi is homeomorphic to X for all large i. 1.
A knot characterization and 1connected nonnegatively curved 4manifolds with circle symmetry
 In arXiv:1304.4827 [math.DG
, 2013
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WASSERSTEIN GEOMETRY OF GAUSSIAN MEASURES
"... Abstract. This paper concerns the Riemannian/Alexandrov geometry of Gaussian measures, from the view point of the L 2Wasserstein geometry. The space of Gaussian measures is of finite dimension, which allows to write down the explicit Riemannian metric which in turn induces the L 2Wasserstein dista ..."
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Cited by 11 (4 self)
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Abstract. This paper concerns the Riemannian/Alexandrov geometry of Gaussian measures, from the view point of the L 2Wasserstein geometry. The space of Gaussian measures is of finite dimension, which allows to write down the explicit Riemannian metric which in turn induces the L 2Wasserstein distance. Moreover, its completion as a metric space provides a complete picture of the singular behavior of the L 2Wasserstein geometry. In particular, the singular set is stratified according to the dimension of the support of the Gaussian measures, providing an explicit nontrivial example of Alexandrov space with extremal sets. 1.
Fréchet means for distributions of persistence diagrams
"... Abstract. Given a distribution ρ on persistence diagrams and observations X1,...Xn iid ∼ ρ we introduce an algorithm in this paper that estimates a Fréchet mean from the set of diagrams X1,...Xn. If the underlying measure ρ is a combination of Dirac masses ρ = 1 m ∑m i=1 δZi then we prove the algo ..."
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Cited by 10 (3 self)
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Abstract. Given a distribution ρ on persistence diagrams and observations X1,...Xn iid ∼ ρ we introduce an algorithm in this paper that estimates a Fréchet mean from the set of diagrams X1,...Xn. If the underlying measure ρ is a combination of Dirac masses ρ = 1 m ∑m i=1 δZi then we prove the algorithm converges to a local minimum and a law of large numbers result for a Fréchet mean computed by the algorithm given observations drawn iid from ρ. We illustrate the convergence of an empirical mean computed by the algorithm to a population mean by simulations from Gaussian random fields. 1.
Yau’s gradient estimates on Alexandrov spaces
 University of Bonn, Institute for
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A SIMPLE PROOF OF PERELMAN’S COLLAPSING THEOREM FOR 3MANIFOLDS
"... Abstract. We will simplify earlier proofs of Perelman’s collapsing theorem for 3manifolds given by ShioyaYamaguchi [SY00][SY05] and MorganTian [MT08]. A version of Perelman’s collapsing theorem states: “Let {M 3 i} be a sequence of compact Riemannian 3manifolds with curvature bounded from below ..."
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Cited by 8 (2 self)
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Abstract. We will simplify earlier proofs of Perelman’s collapsing theorem for 3manifolds given by ShioyaYamaguchi [SY00][SY05] and MorganTian [MT08]. A version of Perelman’s collapsing theorem states: “Let {M 3 i} be a sequence of compact Riemannian 3manifolds with curvature bounded from below by (−1) and diam(M 3 i) ≥ c0> 0. Suppose that all unit metric balls in M 3 i have very small volume at most is closed or it vi → 0 as i → ∞ and suppose that either M 3 i has possibly convex incompressible toral boundary. Then M 3 i must be a graphmanifold for sufficiently large i”. This result can be viewed as an extension of implicit function theorem. Among other things, we use Perelman’s critical point theory (e.g., multiple conic singularity theory and his fibration theory) for Alexandrov spaces to construct the desired local Seifert fibration structure on collapsed 3manifolds. The verification of Perelman’s collapsing theorem is the last step of Perelman’s proof of Thurston’s Geometrization Conjecture on the classification of 3manifolds. Our proof of Perelman’s collapsing theorem is accessible to nonexperts and advanced graduate students. Contents
Orientation and symmetries of Alexandrov spaces with applications in positive curvature
, 2013
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An Optimal Extension of Perelman’s Comparison Theorem for Quadrangles and its Applications
"... In this paper we discuss an extension of Perelman’s comparison for quadrangles. Among applications of this new comparison theorem, we study the equidistance evolution of hypersurfaces in Alexandrov spaces with nonnegative curvature. We show that, in certain cases, the equidistance evolution of hype ..."
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Cited by 7 (4 self)
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In this paper we discuss an extension of Perelman’s comparison for quadrangles. Among applications of this new comparison theorem, we study the equidistance evolution of hypersurfaces in Alexandrov spaces with nonnegative curvature. We show that, in certain cases, the equidistance evolution of hypersurfaces become totally convex relative to a bigger subdomain. An optimal extension of 2nd variational formula for geodesics by Petrunin will be derived for the case of nonnegative curvature. In addition, we also introduced the generalized second fundament forms for subsets in Alexandrov spaces. Using this new notion, we will propose an approach to study two open problems in Alexandrov geometry.
Nilpotency, almost nonnegative curvature, and the gradient flow on Alexandrov spaces,
 Ann. of Math.
, 2010
"... Abstract. We show that almost nonnegatively curved m manifolds are, up to finite cover, nilpotent spaces in the sense of homotopy theory and have C(m) nilpotent fundamental groups. We also show that up to a finite cover almost nonnegatively curved manifolds are fiber bundles with simply connected ..."
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Cited by 6 (2 self)
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Abstract. We show that almost nonnegatively curved m manifolds are, up to finite cover, nilpotent spaces in the sense of homotopy theory and have C(m) nilpotent fundamental groups. We also show that up to a finite cover almost nonnegatively curved manifolds are fiber bundles with simply connected fibers over nilmanifolds.