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Asymptotics for spherical needlets
 The Annals of Statistics
"... We investigate invariant random fields on the sphere using a new type of spherical wavelets, called needlets. These are compactly supported in frequency and enjoy excellent localization properties in real space, with quasiexponentially decaying tails. We show that, for random fields on the sphere, ..."
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Cited by 30 (13 self)
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We investigate invariant random fields on the sphere using a new type of spherical wavelets, called needlets. These are compactly supported in frequency and enjoy excellent localization properties in real space, with quasiexponentially decaying tails. We show that, for random fields on the sphere, the needlet coefficients are asymptotically uncorrelated for any fixed angular distance. This property is used to derive CLT and functional CLT convergence results for polynomial functionals of the needlet coefficients: here the asymptotic theory is considered in the highfrequency sense. Our proposals emerge from strong empirical motivations, especially in connection with the analysis of cosmological data sets.
Aa statistical approach to persistent homology
 Homology Homotopy and Applications
"... Abstract. Assume that a finite set of points is randomly sampled from a subspace of a metric space. Recent advances in computational topology have provided several approaches to recovering the geometric and topological properties of the underlying space. In this paper we take a statistical approach ..."
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Cited by 20 (5 self)
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Abstract. Assume that a finite set of points is randomly sampled from a subspace of a metric space. Recent advances in computational topology have provided several approaches to recovering the geometric and topological properties of the underlying space. In this paper we take a statistical approach to this problem. We assume that the data is randomly sampled from an unknown probability distribution. We define two filtered complexes with which we can calculate the persistent homology of a probability distribution. Using statistical estimators for samples from certain families of distributions, we show that we can recover the persistent homology of the underlying distribution. 1.
Statistical Topology Via Morse Theory Persistence and Nonparametric Estimation
 Algebraic Methods in Statistics and Probability II
, 2010
"... 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 of persistent homology. This estimation procedure can then be evaluated using the bottleneck distance ..."
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Cited by 12 (1 self)
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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 of persistent homology. This estimation procedure can then be evaluated using the bottleneck distance between the estimated persistent homology and the true persistent homology. The connection to statistics comes from the fact that when viewed as a nonparametric regression problem, the bottleneck distance is bounded by the supnorm loss. Consequently, a sharp asymptotic minimax bound is determined under the sup–norm risk over Hölder classes of functions for the nonparametric regression problem on manifolds. This provides good convergence properties for the persistent homology estimator in terms of the expected bottleneck distance.
unknown title
, 704
"... On the characterization of isotropic Gaussian fields on homogeneous spaces of compact groups ..."
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On the characterization of isotropic Gaussian fields on homogeneous spaces of compact groups
Data Analysis using Computational Topology and Geometric Statistics
, 2009
"... Mathematical scientists of diverse backgrounds are being asked to apply the techniques of their specialty to data which is greater in both size and complexity than that which has been studied previously. Large, highdimensional data sets, for which traditional linear methods are inadequate, pose cha ..."
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Mathematical scientists of diverse backgrounds are being asked to apply the techniques of their specialty to data which is greater in both size and complexity than that which has been studied previously. Large, highdimensional data sets, for which traditional linear methods are inadequate, pose challenges in representation, visualization, interpretation and analysis. A common finding is that these massive data sets require the development of new theory and that these advances are dependent on increasing technical sophistication. Two such dataanalytic techniques that have recently developed independently of each other have come to the fore, namely, Geometric Statistics and Computational Topology. Although the former uses geometric arguments, while the latter uses algebraictopological arguments, and hence they appear disparate, there is substantial commonality and overlap just as in the more traditional overlap between geometry and topology. Thus the purpose of this workshop is to bring together these two research directions and explore their overlap, particularly in the service of statistical data analysis. A standard paradigm assumes that the data comes from some underlying geometric structure, such as a curved submanifold or a singular algebraic variety. The observed data is obtained as a random sample from this space, and the objective is to statistically recover features of the underlying space and/or the distribution that generated the sample.
© Institute of Mathematical Statistics, 2009 ASYMPTOTICS FOR SPHERICAL NEEDLETS
"... We investigate invariant random fields on the sphere using a new type of spherical wavelets, called needlets. These are compactly supported in frequency and enjoy excellent localization properties in real space, with quasiexponentially decaying tails. We show that, for random fields on the sphere, ..."
Abstract
 Add to MetaCart
We investigate invariant random fields on the sphere using a new type of spherical wavelets, called needlets. These are compactly supported in frequency and enjoy excellent localization properties in real space, with quasiexponentially decaying tails. We show that, for random fields on the sphere, the needlet coefficients are asymptotically uncorrelated for any fixed angular distance. This property is used to derive CLT and functional CLT convergence results for polynomial functionals of the needlet coefficients: here the asymptotic theory is considered in the highfrequency sense. Our proposals emerge from strong empirical motivations, especially in connection with the analysis of cosmological data sets.