Topology of random geometric complexes: a survey
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paper_title: Homological connectivity of random k-dimensional complexes

paper_content:

Let Δn-1 denote the (n - 1)-dimensional simplex. Let Y be a random k-dimensional subcomplex of Δn-1 obtained by starting with the full (k - 1)-dimensional skeleton of Δn-1 and then adding each k-simplex independently with probability p. Let Hk-1(Y; R) denote the (k - 1)-dimensional reduced homology group of Y with coefficients in a finite abelian group R. It is shown that for any fixed R and k ≥ 1 and for any function ω(n) that tends to infinity $${\mathop{\rm lim}\limits_{n \rightarrow \infty}}{\Pr}[H_{k-1} \; (Y;R)= 0]=\left\{ \eqalign {&{0 \quad p= {k\; {\rm log}\;n-\omega(n) \over n }} \cr & 1 \quad p={k\; {\rm log}\; n + \omega (n) \over n}. } \right. $$ © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009
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paper_title: Homological Connectivity of Random 2-Complexes

paper_content:

Let Δn−1 denote the (n − 1)-dimensional simplex. Let Y be a random 2-dimensional subcomplex of Δn−1 obtained by starting with the full 1-dimensional skeleton of Δn−1 and then adding each 2−simplex independently with probability p. Let $$H_{1} {\left( {Y;{\Bbb F}_{2} } \right)}$$ denote the first homology group of Y with mod 2 coefficients. It is shown that for any function ω(n) that tends to infinity $${\mathop {\lim }\limits_{n \to \infty } }{\kern 1pt} {\kern 1pt} {\text{Prob}}{\left[ {H_{1} {\left( {Y;{\Bbb F}_{2} } \right)} = 0} \right]} = \left\{ {\begin{array}{*{20}c}{{0p = \frac{{2\log n - \omega {\left( n \right)}}}{n}}} \\ {{1p = \frac{{2\log n + \omega {\left( n \right)}}}{n}}} \\ \end{array} } \right.$$
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paper_title: Topological consistency via kernel estimation

paper_content:

We introduce a consistent estimator for the homology (an algebraic structure representing connected components and cycles) of level 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 multi-scale 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.
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paper_title: Random Geometric Graphs

paper_content:

1. Introduction 2. Probabilistic ingredients 3. Subgraph and component counts 4. Typical vertex degrees 5. Geometrical ingredients 6. Maximum degree, cliques and colourings 7. Minimum degree: laws of large numbers 8. Minimum degree: convergence in distribution 9. Percolative ingredients 10. Percolation and the largest component 11. The largest component for a binomial process 12. Ordering and partitioning problems 13. Connectivity and the number of components References Index
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paper_title: The fundamental group of random 2-complexes

paper_content:

We study Linial-Meshulam random 2-complexes, which are two-dimensional analogues of Erd\H{o}s-R\'enyi random graphs. We find the threshold for simple connectivity to be p = n^{-1/2}. This is in contrast to the threshold for vanishing of the first homology group, which was shown earlier by Linial and Meshulam to be p = 2 log(n)/n. We use a variant of Gromov's local-to-global theorem for linear isoperimetric inequalities to show that when p = O(n^{-1/2 -\epsilon}) the fundamental group is word hyperbolic. Along the way we classify the homotopy types of sparse 2-dimensional simplicial complexes and establish isoperimetric inequalities for such complexes. These intermediate results do not involve randomness and may be of independent interest.
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paper_title: Tight Lower Bounds for Homology Inference

paper_content:

The homology groups of a manifold are important topological invariants that provide an algebraic summary of the manifold. These groups contain rich topological information, for instance, about the connected components, holes, tunnels and sometimes the dimension of the manifold. In earlier work, we have considered the statistical problem of estimating the homology of a manifold from noiseless samples and from noisy samples under several different noise models. We derived upper and lower bounds on the minimax risk for this problem. In this note we revisit the noiseless case. In previous work we used Le Cam's lemma to establish a lower bound that differed from the upper bound of Niyogi, Smale and Weinberger by a polynomial factor in the condition number. In this note we use a different construction based on the direct analysis of the likelihood ratio test to show that the upper bound of Niyogi, Smale and Weinberger is in fact tight, thus establishing rate optimal asymptotic minimax bounds for the problem. The techniques we use here extend in a straightforward way to the noisy settings considered in our earlier work.
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paper_title: Barcodes: The persistent topology of data

paper_content:

This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The primary mathematical tool considered is a homology theory for point-cloud data sets — persistent homology — and a novel representation of this algebraic characterization — barcodes. We sketch an application of these techniques to the classification of natural images. 1. The shape of data When a topologist is asked, “How do you visualize a four-dimensional object?” the appropriate response is a Socratic rejoinder: “How do you visualize a threedimensional object?” We do not see in three spatial dimensions directly, but rather via sequences of planar projections integrated in a manner that is sensed if not comprehended. We spend a significant portion of our first year of life learning how to infer three-dimensional spatial data from paired planar projections. Years of practice have tuned a remarkable ability to extract global structure from representations in a strictly lower dimension. The inference of global structure occurs on much finer scales as well, with regards to converting discrete data into continuous images. Dot-matrix printers, scrolling LED tickers, televisions, and computer displays all communicate images via arrays of discrete points which are integrated into coherent, global objects. This also is a skill we have practiced from childhood. No adult does a dot-to-dot puzzle with anything approaching anticipation. 1.1. Topological data analysis. Problems of data analysis share many features with these two fundamental integration tasks: (1) how does one infer high dimensional structure from low dimensional representations; and (2) how does one assemble discrete points into global structure. The principal themes of this survey of the work of Carlsson, de Silva, Edelsbrunner, Harer, Zomorodian, and others are the following: (1) It is beneficial to replace a set of data points with a family of simplicial complexes, indexed by a proximity parameter. This converts the data set into global topological objects. (2) It is beneficial to view these topological complexes through the lens of algebraic topology — specifically, via a novel theory of persistent homology adapted to parameterized families. (3) It is beneficial to encode the persistent homology of a data set in the form of a parameterized version of a Betti number: a barcode. The author gratefully acknowledges the support of DARPA # HR0011-07-1-0002. The work reviewed in this article is funded by the DARPA program TDA: Topological Data Analysis.
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paper_title: Finding the homology of submanifolds with high confidence from random samples

paper_content:

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 are 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 an algorithm to do this and provide learning-theoretic complexity bounds. Our bounds are obtained in terms of a condition number that limits the curvature and nearness to self-intersection of the submanifold. We are also able to treat the situation where the data are “noisy” and lie near rather than on the submanifold in question.
---
paper_title: A topological view of unsupervised learning from noisy data

paper_content:

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 any underlying geometrically structured probability distribution in a certain sense that we will make precise. We construct a geometrically structured probability distribution that seems appropriate for modeling data in very high dimensions. A special case of our construction is the mixture of Gaussians where there is Gaussian noise concentrated around a finite set of points (the means). More generally we consider Gaussian noise concentrated around a low dimensional manifold and discuss how to recover the homology of this underlying geometric core from data that do not lie on it. We show that if the variance of the Gaussian noise is small in a certain sense, then the homology can be learned with high confidence by an algorithm that has a weak (linear) dependence on the ambient dimension. Our algorithm has a natural interpretation as a spectral learning algorithm using a combinatorial Laplacian of a suitable data-derived simplicial complex.
---
paper_title: On the topology of random complexes built over stationary point processes

paper_content:

Cech and Vietoris-Rips complexes. In particular, we obtain results about homology, as measured via the growth of Betti numbers, when the vertices are the points of a general stationary point process. This significantly extends earlier results in which the points were either iid observations or the points of a Poisson process. In dealing with general point processes, in which the points exhibit dependence such as attraction, or repulsion, we find phenomena quantitatively different from those observed in the iid and Poisson cases. From the point of view of topological data analysis, our results seriously impact on considerations of model (non) robustness for statistical inference.
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paper_title: Elements of Algebraic Topology

paper_content:

Elements of Algebraic Topology provides the most concrete approach to the subject. With coverage of homology and cohomology theory, universal coefficient theorems, Kunneth theorem, duality in manifolds, and applications to classical theorems of point-set topology, this book is perfect for comunicating complex topics and the fun nature of algebraic topology for beginners.
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paper_title: Random Geometric Graphs

paper_content:

1. Introduction 2. Probabilistic ingredients 3. Subgraph and component counts 4. Typical vertex degrees 5. Geometrical ingredients 6. Maximum degree, cliques and colourings 7. Minimum degree: laws of large numbers 8. Minimum degree: convergence in distribution 9. Percolative ingredients 10. Percolation and the largest component 11. The largest component for a binomial process 12. Ordering and partitioning problems 13. Connectivity and the number of components References Index
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paper_title: Size of the Giant Component in a Random Geometric Graph

paper_content:

In this paper, we study the size of the giant component CG in the random geometric graph G = G(n, rn, f) of n nodes independently distributed each according to a certain density f(.) in [0, 1]2 satisfying infx∈[0,1]2 f(x) > 0. If c1 n ≤ r 2 n ≤ c2 logn n for some positive constants c1, c2 and nr 2 n −→ ∞, we show that the giant component of G contains at least n − o(n) nodes with probability at least 1 − o(1) as n → ∞. We also obtain estimates on the diameter and number of the non-giant components of G.
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paper_title: Limit theorems for Betti numbers of random simplicial complexes

paper_content:

There have been several recent articles studying homology of various types of random simplicial complexes. Several theorems have concerned thresholds for vanishing of homology, and in some cases expectations of the Betti numbers. However little seems known so far about limiting distributions of random Betti numbers. ::: In this article we establish Poisson and normal approximation theorems for Betti numbers of different kinds of random simplicial complex: Erd\H{o}s-R\'enyi random clique complexes, random Vietoris-Rips complexes, and random \v{C}ech complexes. These results may be of practical interest in topological data analysis.
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paper_title: Random geometric complexes in the thermodynamic regime

paper_content:

We consider the topology of simplicial complexes with vertices the points of a random point process and faces determined by distance relationships between the vertices. In particular, we study the Betti numbers of these complexes as the number of vertices becomes large, obtaining limit theorems for means, strong laws, concentration inequalities and central limit theorems. As opposed to most prior papers treating random complexes, the limit with which we work is in the so-called ‘thermodynamic’ regime (which includes the percolation threshold) in which the complexes become very large and complicated, with complex homology characterised by diverging Betti numbers. The proofs combine probabilistic arguments from the theory of stabilizing functionals of point processes and topological arguments exploiting the properties of Mayer–Vietoris exact sequences. The Mayer–Vietoris arguments are crucial, since homology in general, and Betti numbers in particular, are global rather than local phenomena, and most standard probabilistic arguments are based on the additivity of functionals arising as a consequence of locality.
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paper_title: Betti number signatures of homogeneous Poisson point processes

paper_content:

The Betti numbers are fundamental topological quantities that describe the k-dimensional connectivity of an object: B_0 is the number of connected components and B_k effectively counts the number of k-dimensional holes. Although they are appealing natural descriptors of shape, the higher-order Betti numbers are more difficult to compute than other measures and so have not previously been studied per se in the context of stochastic geometry or statistical physics. ::: As a mathematically tractable model, we consider the expected Betti numbers per unit volume of Poisson-centred spheres with radius alpha. We present results from simulations and derive analytic expressions for the low intensity, small radius limits of Betti numbers in one, two, and three dimensions. The algorithms and analysis depend on alpha-shapes, a construction from computational geometry that deserves to be more widely known in the physics community.
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paper_title: On the Shape of a Set of Points in the Plane

paper_content:

A generalization of the convex hull of a finite set of points in the plane is introduced and analyzed. This generalization leads to a family of straight-line graphs, " \alpha -shapes," which seem to capture the intuitive notions of "fine shape" and "crude shape" of point sets. It is shown that a-shapes are subgraphs of the closest point or furthest point Delaunay triangulation. Relying on this result an optimal O(n \log n) algorithm that constructs \alpha -shapes is developed.
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paper_title: Random geometric complexes

paper_content:

We study the expected topological properties of Cech and Vietoris–Rips complexes built on random points in ℝd . We find higher-dimensional analogues of known results for connectivity and component counts for random geometric graphs. However, higher homology H k is not monotone when k>0. ::: ::: In particular, for every k>0, we exhibit two thresholds, one where homology passes from vanishing to nonvanishing, and another where it passes back to vanishing. We give asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes.
---
paper_title: On the topology of random complexes built over stationary point processes

paper_content:

Cech and Vietoris-Rips complexes. In particular, we obtain results about homology, as measured via the growth of Betti numbers, when the vertices are the points of a general stationary point process. This significantly extends earlier results in which the points were either iid observations or the points of a Poisson process. In dealing with general point processes, in which the points exhibit dependence such as attraction, or repulsion, we find phenomena quantitatively different from those observed in the iid and Poisson cases. From the point of view of topological data analysis, our results seriously impact on considerations of model (non) robustness for statistical inference.
---
paper_title: Random geometric complexes in the thermodynamic regime

paper_content:

We consider the topology of simplicial complexes with vertices the points of a random point process and faces determined by distance relationships between the vertices. In particular, we study the Betti numbers of these complexes as the number of vertices becomes large, obtaining limit theorems for means, strong laws, concentration inequalities and central limit theorems. As opposed to most prior papers treating random complexes, the limit with which we work is in the so-called ‘thermodynamic’ regime (which includes the percolation threshold) in which the complexes become very large and complicated, with complex homology characterised by diverging Betti numbers. The proofs combine probabilistic arguments from the theory of stabilizing functionals of point processes and topological arguments exploiting the properties of Mayer–Vietoris exact sequences. The Mayer–Vietoris arguments are crucial, since homology in general, and Betti numbers in particular, are global rather than local phenomena, and most standard probabilistic arguments are based on the additivity of functionals arising as a consequence of locality.
---
paper_title: Random geometric complexes

paper_content:

We study the expected topological properties of Cech and Vietoris–Rips complexes built on random points in ℝd . We find higher-dimensional analogues of known results for connectivity and component counts for random geometric graphs. However, higher homology H k is not monotone when k>0. ::: ::: In particular, for every k>0, we exhibit two thresholds, one where homology passes from vanishing to nonvanishing, and another where it passes back to vanishing. We give asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes.
---
paper_title: Random Geometric Graphs

paper_content:

1. Introduction 2. Probabilistic ingredients 3. Subgraph and component counts 4. Typical vertex degrees 5. Geometrical ingredients 6. Maximum degree, cliques and colourings 7. Minimum degree: laws of large numbers 8. Minimum degree: convergence in distribution 9. Percolative ingredients 10. Percolation and the largest component 11. The largest component for a binomial process 12. Ordering and partitioning problems 13. Connectivity and the number of components References Index
---
paper_title: Finding the homology of submanifolds with high confidence from random samples

paper_content:

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 are 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 an algorithm to do this and provide learning-theoretic complexity bounds. Our bounds are obtained in terms of a condition number that limits the curvature and nearness to self-intersection of the submanifold. We are also able to treat the situation where the data are “noisy” and lie near rather than on the submanifold in question.
---
paper_title: A topological view of unsupervised learning from noisy data

paper_content:

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 any underlying geometrically structured probability distribution in a certain sense that we will make precise. We construct a geometrically structured probability distribution that seems appropriate for modeling data in very high dimensions. A special case of our construction is the mixture of Gaussians where there is Gaussian noise concentrated around a finite set of points (the means). More generally we consider Gaussian noise concentrated around a low dimensional manifold and discuss how to recover the homology of this underlying geometric core from data that do not lie on it. We show that if the variance of the Gaussian noise is small in a certain sense, then the homology can be learned with high confidence by an algorithm that has a weak (linear) dependence on the ambient dimension. Our algorithm has a natural interpretation as a spectral learning algorithm using a combinatorial Laplacian of a suitable data-derived simplicial complex.
---
paper_title: Random geometric complexes in the thermodynamic regime

paper_content:

We consider the topology of simplicial complexes with vertices the points of a random point process and faces determined by distance relationships between the vertices. In particular, we study the Betti numbers of these complexes as the number of vertices becomes large, obtaining limit theorems for means, strong laws, concentration inequalities and central limit theorems. As opposed to most prior papers treating random complexes, the limit with which we work is in the so-called ‘thermodynamic’ regime (which includes the percolation threshold) in which the complexes become very large and complicated, with complex homology characterised by diverging Betti numbers. The proofs combine probabilistic arguments from the theory of stabilizing functionals of point processes and topological arguments exploiting the properties of Mayer–Vietoris exact sequences. The Mayer–Vietoris arguments are crucial, since homology in general, and Betti numbers in particular, are global rather than local phenomena, and most standard probabilistic arguments are based on the additivity of functionals arising as a consequence of locality.
---
paper_title: On the topology of random complexes built over stationary point processes

paper_content:

Cech and Vietoris-Rips complexes. In particular, we obtain results about homology, as measured via the growth of Betti numbers, when the vertices are the points of a general stationary point process. This significantly extends earlier results in which the points were either iid observations or the points of a Poisson process. In dealing with general point processes, in which the points exhibit dependence such as attraction, or repulsion, we find phenomena quantitatively different from those observed in the iid and Poisson cases. From the point of view of topological data analysis, our results seriously impact on considerations of model (non) robustness for statistical inference.
---
paper_title: Barcodes: The persistent topology of data

paper_content:

This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The primary mathematical tool considered is a homology theory for point-cloud data sets — persistent homology — and a novel representation of this algebraic characterization — barcodes. We sketch an application of these techniques to the classification of natural images. 1. The shape of data When a topologist is asked, “How do you visualize a four-dimensional object?” the appropriate response is a Socratic rejoinder: “How do you visualize a threedimensional object?” We do not see in three spatial dimensions directly, but rather via sequences of planar projections integrated in a manner that is sensed if not comprehended. We spend a significant portion of our first year of life learning how to infer three-dimensional spatial data from paired planar projections. Years of practice have tuned a remarkable ability to extract global structure from representations in a strictly lower dimension. The inference of global structure occurs on much finer scales as well, with regards to converting discrete data into continuous images. Dot-matrix printers, scrolling LED tickers, televisions, and computer displays all communicate images via arrays of discrete points which are integrated into coherent, global objects. This also is a skill we have practiced from childhood. No adult does a dot-to-dot puzzle with anything approaching anticipation. 1.1. Topological data analysis. Problems of data analysis share many features with these two fundamental integration tasks: (1) how does one infer high dimensional structure from low dimensional representations; and (2) how does one assemble discrete points into global structure. The principal themes of this survey of the work of Carlsson, de Silva, Edelsbrunner, Harer, Zomorodian, and others are the following: (1) It is beneficial to replace a set of data points with a family of simplicial complexes, indexed by a proximity parameter. This converts the data set into global topological objects. (2) It is beneficial to view these topological complexes through the lens of algebraic topology — specifically, via a novel theory of persistent homology adapted to parameterized families. (3) It is beneficial to encode the persistent homology of a data set in the form of a parameterized version of a Betti number: a barcode. The author gratefully acknowledges the support of DARPA # HR0011-07-1-0002. The work reviewed in this article is funded by the DARPA program TDA: Topological Data Analysis.
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paper_title: Aa statistical approach to persistent homology

paper_content:

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.
---
paper_title: Random geometric complexes in the thermodynamic regime

paper_content:

We consider the topology of simplicial complexes with vertices the points of a random point process and faces determined by distance relationships between the vertices. In particular, we study the Betti numbers of these complexes as the number of vertices becomes large, obtaining limit theorems for means, strong laws, concentration inequalities and central limit theorems. As opposed to most prior papers treating random complexes, the limit with which we work is in the so-called ‘thermodynamic’ regime (which includes the percolation threshold) in which the complexes become very large and complicated, with complex homology characterised by diverging Betti numbers. The proofs combine probabilistic arguments from the theory of stabilizing functionals of point processes and topological arguments exploiting the properties of Mayer–Vietoris exact sequences. The Mayer–Vietoris arguments are crucial, since homology in general, and Betti numbers in particular, are global rather than local phenomena, and most standard probabilistic arguments are based on the additivity of functionals arising as a consequence of locality.
---
paper_title: Topology of random simplicial complexes: a survey

paper_content:

This expository article is based on a lecture from the Stanford Symposium on Algebraic Topology: Application and New Directions, held in honor of Gunnar Carlsson, Ralph Cohen, and Ib Madsen.
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Title: Topology of random geometric complexes: a survey

Section 1: Introduction
Description 1: Overview of the paper’s purpose and a brief introduction to the topic and its significance.

Section 2: Preliminaries
Description 2: Introduce the concepts and notation that will be used throughout the survey.

Section 3: Connectivity
Description 3: Review the classical results about the connectivity of random geometric graphs, and their implications on simplicial complexes.

Section 4: Homology and Betti Numbers
Description 4: Discuss the main results known to date about the limiting behavior of the homology of random geometric complexes, including the subcritical, critical, and supercritical regimes.

Section 5: Morse Theory for the Distance Function
Description 5: Review an alternative approach to study the homology of random Čech complexes using Morse theory for the distance function.

Section 6: Extensions to Manifolds
Description 6: Explore extensions to the results in Section 4 for compact manifolds and their implications.

Section 7: Stationary Point Processes
Description 7: Review recent results extending random geometric complex frameworks to more general stationary point processes.

Section 8: Persistent Homology
Description 8: Discuss work in progress related to persistent homology generated by random geometric complexes and its significance in data analysis.

Section 9: Open Problems and Future Directions
Description 9: Present a list of open problems and possible future research areas in the field.