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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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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.
Metric graph reconstruction from noisy data
 IN PROC. 27TH SYMPOS. COMPUT. GEOM
, 2011
"... Many realworld data sets can be viewed of as noisy samples of special types of metric spaces called metric graphs [16]. Building on the notions of correspondence and GromovHausdorff distance in metric geometry, we describe a model for such data sets as an approximation of an underlying metric grap ..."
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Cited by 9 (1 self)
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Many realworld data sets can be viewed of as noisy samples of special types of metric spaces called metric graphs [16]. Building on the notions of correspondence and GromovHausdorff distance in metric geometry, we describe a model for such data sets as an approximation of an underlying metric graph. We present a novel algorithm that takes as an input such a data set, and outputs the underlying metric graph with guarantees. We also implement the algorithm, and evaluate its performance on a variety of real world data sets.
Data Skeletonization via Reeb Graphs
"... Recovering hidden structure from complex and noisy nonlinear data is one of the most fundamental problems in machine learning and statistical inference. While such data is often highdimensional, it is of interest to approximate it with a lowdimensional or even onedimensional space, since many im ..."
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Recovering hidden structure from complex and noisy nonlinear data is one of the most fundamental problems in machine learning and statistical inference. While such data is often highdimensional, it is of interest to approximate it with a lowdimensional or even onedimensional space, since many important aspects of data are often intrinsically lowdimensional. Furthermore, there are many scenarios where the underlying structure is graphlike, e.g, river/road networks or various trajectories. In this paper, we develop a framework to extract, as well as to simplify, a onedimensional ”skeleton ” from unorganized data using the Reeb graph. Our algorithm is very simple, does not require complex optimizations and can be easily applied to unorganized highdimensional data such as point clouds or proximity graphs. It can also represent arbitrary graph structures in the data. We also give theoretical results to justify our method. We provide a number of experiments to demonstrate the effectiveness and generality of our algorithm, including comparisons to existing methods, such as principal curves. We believe that the simplicity and practicality of our algorithm will help to promote skeleton graphs as a data analysis tool for a broad range of applications. 1
The topology of probability distributions on manifolds
, 2014
"... Let P be a set of n random points in Rd, generated from a probability measure on a mdimensional manifold M ⊂ Rd. In this paper we study the homology of U(P, r) – the union of ddimensional balls of radius r around P, as n→∞, and r → 0. In addition we study the critical points of dP – the distance ..."
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Let P be a set of n random points in Rd, generated from a probability measure on a mdimensional manifold M ⊂ Rd. In this paper we study the homology of U(P, r) – the union of ddimensional balls of radius r around P, as n→∞, and r → 0. In addition we study the critical points of dP – the distance function from the set P. These two objects are known to be related via Morse theory. We present limit theorems for the Betti numbers of U(P, r), as well as for number of critical points of index k for dP. Depending on how fast r decays to zero as n grows, these two objects exhibit different types of limiting behavior. In one particular case (nrm ≥ C logn), we show that the Betti numbers of U(P, r) perfectly recover the Betti numbers of the original manifold M, a result which is of significant interest in topological manifold learning.
Approximating Local Homology from Samples
"... Abstract. Recently, multiscale notions of local homology (a variant of persistent homology) have been used to study the local structure of spaces around a given point from a point cloud sample. Current reconstruction guarantees rely on constructing embedded complexes which become difficult in high ..."
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Abstract. Recently, multiscale notions of local homology (a variant of persistent homology) have been used to study the local structure of spaces around a given point from a point cloud sample. Current reconstruction guarantees rely on constructing embedded complexes which become difficult in high dimensions. We show that the persistence diagrams used for estimating local homology, can be approximated using families of VietorisRips complexes, whose simple constructions are robust in any dimension. To the best of our knowledge, our results, for the first time, make applications based on local homology, such as stratification learning, feasible in high dimensions. 1
TOPOLOGICAL CONSISTENCY VIA KERNEL ESTIMATION
, 2014
"... We introduce a consistent estimator for the homology (an algebraic structure representing connected components and cycles) of superlevel sets of both density and regression functions. Our method is based on kernel estimation. We apply this procedure to two problems: 1) inferring the homology struc ..."
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We introduce a consistent estimator for the homology (an algebraic structure representing connected components and cycles) of superlevel sets of both density and regression functions. Our method is based on kernel estimation. We apply this procedure to two problems: 1) inferring the homology structure of manifolds from noisy observations, 2) inferring the persistent homology (a multiscale extension of homology) of either density or regression functions. We prove consistency for both of these problems. In addition to the theoretical results we demonstrate these methods on simulated data for binary regression and clustering applications.
Statistical Analysis of Metric Graph Reconstruction
"... A metric graph is a 1dimensional stratified metric space consisting of vertices and edges or loops glued together. Metric graphs can be naturally used to represent and model data that take the form of noisy filamentary structures, such as street maps, neurons, networks of rivers and galaxies. We co ..."
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A metric graph is a 1dimensional stratified metric space consisting of vertices and edges or loops glued together. Metric graphs can be naturally used to represent and model data that take the form of noisy filamentary structures, such as street maps, neurons, networks of rivers and galaxies. We consider the statistical problem of reconstructing the topology of a metric graph from a random sample. We derive a lower bound on the minimax risk for the noiseless case and an upper bound for the special case of metric graphs embedded in R2. The upper bound is based on the reconstruction algorithm given in Aanjaneya et al. (2012).
Toward Understanding Complex Spaces: Graph Laplacians on Manifolds with Singularities and Boundaries
, 2012
"... In manifold learning, algorithms based on graph Laplacian constructed from data have received considerable attention both in practical applications and theoretical analysis. Much of the existing work has been done under the assumption that the data is sampled from a manifold without boundaries and s ..."
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In manifold learning, algorithms based on graph Laplacian constructed from data have received considerable attention both in practical applications and theoretical analysis. Much of the existing work has been done under the assumption that the data is sampled from a manifold without boundaries and singularities or that the functions of interest are evaluated away from such points. At the same time, it can be argued that singularities and boundaries are an important aspect of the geometry of realistic data. Boundaries occur whenever the process generating data has a bounding constraint; while singularities appear when two different manifolds intersect or if a process undergoes a “phase transition”, changing nonsmoothly as a function of a parameter. In this paper we consider the behavior of graph Laplacians at points at or near boundaries and two main types of other singularities: intersections, where different manifolds come together and sharp “edges”, where a manifold sharply changes direction. We show that the behavior of graph Laplacian near these singularities is quite different from that in the interior of the manifolds. In fact, a phenomenon somewhat reminiscent of the Gibbs effect in the analysis of Fourier series, can be observed in the behavior of graph Laplacian near such points. Unlike in the interior of the domain,