Results 1  10
of
29
Finding the homology of submanifolds with high confidence from random samples
, 2004
"... 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 hom ..."
Abstract

Cited by 117 (7 self)
 Add to MetaCart
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 an algorithm to do this and provide learningtheoretic complexity bounds. Our bounds are obtained in terms of a condition number that limits the curvature and nearness to selfintersection of the submanifold. We are also able to treat the situation where the data is “noisy ” and lies near rather than on the submanifold in question.
Persistent Homology  a Survey
 CONTEMPORARY MATHEMATICS
"... Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multiscale organization we frequently observe in nature into a mathematical formalism. Here we give a record of the short history of persistent homology and present its basic concepts. ..."
Abstract

Cited by 36 (1 self)
 Add to MetaCart
Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multiscale organization we frequently observe in nature into a mathematical formalism. Here we give a record of the short history of persistent homology and present its basic concepts. Besides the mathematics we focus on algorithms and mention the various connections to applications, including to biomolecules, biological networks, data analysis, and geometric modeling.
Topological Techniques for Efficient Rigorous Computations in Dynamics
, 2001
"... This paper is an expository article on using topological methods for the efficient, rigorous computation of dynamical systems. Of course, since its inception the computer has been used for the purpose of simulating nonlinear models. However, in recent years there has been a rapid development in nume ..."
Abstract

Cited by 27 (10 self)
 Add to MetaCart
This paper is an expository article on using topological methods for the efficient, rigorous computation of dynamical systems. Of course, since its inception the computer has been used for the purpose of simulating nonlinear models. However, in recent years there has been a rapid development in numerical methods specifically designed to study of these models from a dynamical systems point of view, i.e. with a particular emphasis on the structures which capture the longterm or asymptotic states of the system. At the risk of greatly simplifying these results, this work has followed two themes: indirect methods and direct methods. The indirect methods are most closely associated with simulations and as such are extremely important because they tend to be the cheapest computationally. The emphasis is on developing numerical schemes whose solutions exhibit the same dynamics as the original system, e.g. if one is given a Hamiltonian system, then it is reasonable to want a numerical method that preserves the integrals of the original system. A comprehensive introduction to these questions can be found in [61]. The direct methods focus on the development of numerical techniques that find particular dynamical structures, e.g. fixed points, periodic orbits, heteroclinic orbits, invariant manifolds, etc., and are often associated with continuation methods (see [7, 15, 14] and references therein). To paraphrase Poincare, these techniques provide us with a window into the rich structures that nonlinear systems exhibit. There is no question that these methods are essential. However, they cannot capture the full dynamics. As pointed out in [61, p. xiii] a fundamental question for the indirect method, that requires a positive answer, is "Assume that the differential equation has a parti...
A topological view of unsupervised learning from noisy data
 SIAM Journal of Computing
, 2011
"... Abstract. 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. ..."
Abstract

Cited by 14 (0 self)
 Add to MetaCart
Abstract. 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 dataderived simplicial complex.
The tidy set: A minimal simplicial set for computing homology of clique complexes
 In Proc. ACM Symposium of Computational Geometry
, 2010
"... We introduce the tidy set, a minimal simplicial set that captures the topology of a simplicial complex. The tidy set is particularly effective for computing the homology of clique complexes. This family of complexes include the VietorisRips complex and the weak witness complex, methods that are pop ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
We introduce the tidy set, a minimal simplicial set that captures the topology of a simplicial complex. The tidy set is particularly effective for computing the homology of clique complexes. This family of complexes include the VietorisRips complex and the weak witness complex, methods that are popular in topological data analysis. The key feature of our approach is that it skips constructing the clique complex. We give algorithms for constructing tidy sets, implement them, and present experiments. Our preliminary results show that tidy sets are orders of magnitude smaller than clique complexes, giving us a homology engine with small memory requirements.
Spanning trees and Khovanov homology
, 2008
"... The Jones polynomial can be expressed in terms of spanning trees of the graph obtained by checkerboard coloring a knot diagram. We show there exists a complex generated by these spanning trees whose homology is the reduced Khovanov homology. This spanning tree complex is a deformation retract of the ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
The Jones polynomial can be expressed in terms of spanning trees of the graph obtained by checkerboard coloring a knot diagram. We show there exists a complex generated by these spanning trees whose homology is the reduced Khovanov homology. This spanning tree complex is a deformation retract of the reduced Khovanov complex. The spanning trees provide a filtration on the reduced Khovanov complex and a spectral sequence that converges to its homology. For alternating links, the spanning tree complex is the simplest possible because all differentials are zero. Also, the reduced Khovanov homology of a kalmost alternating link lies on at most k+1 adjacent lines. We prove analogous theorems for (unreduced) Khovanov homology. 1
Beyond Graphs: Capturing Groups in Networks
"... Abstract—Currently, the de facto representational choice for networks is graphs. A graph captures pairwise relationships (edges) between entities (vertices) in a network. Network science, however, is replete with group relationships that are more than the sum of the pairwise relationships. For examp ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
Abstract—Currently, the de facto representational choice for networks is graphs. A graph captures pairwise relationships (edges) between entities (vertices) in a network. Network science, however, is replete with group relationships that are more than the sum of the pairwise relationships. For example, collaborative teams, wireless broadcast, insurgent cells, coalitions all contain unique group dynamics that need to be captured in their respective networks. We propose the use of the (abstract) simplicial complex to model groups in networks. We show that a number of problems within social and communications networks such as networkwide broadcast and collaborative teams can be elegantly captured using simplicial complexes in a way that is not possible with graphs. We formulate combinatorial optimization problems in these areas in a simplicial setting and illustrate the applicability of topological concepts such as “Betti numbers ” in structural analysis. As an illustrative case study, we present an analysis of a realworld collaboration network, namely the ARL NSCTA network of researchers and tasks. I.
Curvaturedriven modeling and rendering of pointbased surfaces
 In Braz. Symp. Comp. Graph. Imag. Proc
, 2002
"... In this work we address the problem of computing pointbased surface approximations from point clouds. Our approach is based on recently presented methods that define the approximated surface as the set of stationary points for an operator that projects points in the space onto the surface. We presen ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
In this work we address the problem of computing pointbased surface approximations from point clouds. Our approach is based on recently presented methods that define the approximated surface as the set of stationary points for an operator that projects points in the space onto the surface. We present a novel projection operator that differs from the defined in previous work in that it uses principal curvatures and directions approximation and an anisotropic diffusion equation to ensure an accurate approximation to the surface. We show how to estimate the principal curvatures and directions for point clouds and discuss the usefulness of the curvature information in the context of pointbased surface modeling and rendering. 1
Approximating cycles in a shortest basis of the first homology group from point data. Inverse Problems
, 2012
"... Inference of topological and geometric attributes of a hidden manifold from its point data is a fundamental problem arising in many scientific studies and engineering applications. In this paper we present an algorithm to compute a set of cycles from a point data that presumably sample a smooth mani ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
Inference of topological and geometric attributes of a hidden manifold from its point data is a fundamental problem arising in many scientific studies and engineering applications. In this paper we present an algorithm to compute a set of cycles from a point data that presumably sample a smooth manifold M ⊂ R d. These cycles approximate a shortest basis of the first homology group H1(M) over coefficients in finite field Z2. Previous results addressed the issue of computing the rank of the homology groups from point data, but there is no result on approximating the shortest basis of a manifold from its point sample. In arriving our result, we also present a polynomial time algorithm for computing a shortest basis of H1(K) for any finite simplicial complex K whose edges have nonnegative weights.
Applications of sheaf cohomology and exact sequences to network coding, preprint
, 2011
"... Abstract—Sheaf cohomology is a mathematical tool for collating local algebraic data into global structures. The purpose of this paper is to apply sheaf theory into network coding problems. After the definition of sheaves, we define so called network coding sheaves for a general multi source network ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Abstract—Sheaf cohomology is a mathematical tool for collating local algebraic data into global structures. The purpose of this paper is to apply sheaf theory into network coding problems. After the definition of sheaves, we define so called network coding sheaves for a general multi source network coding scenario, and consider various forms of sheaf cohomologies. The main theorem states that 0th network coding sheaf cohomology is equivalent to information flows for the network coding. Then, this theorem is applied to several practical problems in network codings such as maxflow bounds, global extendability, network robustness, and data merging, by using some of the standard exact sequences of homological algebra. I.