Topological estimation using witness complexes

SIAM Conference on Geometric and Physical Modeling (GD/SPM 2011), 2011

We propose a new iterative algorithm for computing the homology of arbitrary shapes discretized through simplicial complexes. We demonstrate how the simplicial homology of a shape can be effectively expressed in terms of the homology of its sub-components. The proposed algorithm retrieves the complete homological information of an input shape including the Betti numbers, the torsion coefficients and the representative homology generators. To the best of our knowledge, this is the first algorithm based on the constructive Mayer-Vietoris sequence, which relates the homology of a topological space to the homologies of its sub-spaces, i.e. the sub-components of the input shape and their intersections. We demonstrate the validity of our approach through a specific shape decomposition, based only on topological properties, which minimizes the size of the intersections between the sub-components and increases the efficiency of the algorithm.

2003

In this paper, we initiate a study of shape description and classification through the use of persistent homology and three tangential constructions. The homology of our first construction, the tangent complex, can distinguish between topologically identical shapes with different "hard" features, such as sharp corners. To capture "soft" curvature-dependent features, we define two other complexes, the filtered and tame complex. The first is a parametrized family of increasing subcomplexes of the tangent complex. Applying persistent homology, we obtain a shape descriptor in terms of a finite union of intervals. We define a metric over the space of such intervals, arriving at a continuous invariant that reflects the geometric properties of shapes. We illustrate the power of our methods through numerous detailed studies of parametrized families of mathematical shapes. In a later paper, we shall apply our techniques to point cloud data to obtain a computational method of shape recognition based on persistent homology.

Given a point-cloud dataset sampled from an underlying geometric space X, it is often desirable to build a simplicial complex S approximating the geometric or topological structure of X. For example, recent techniques in automatic feature location depend on the ability to estimate topological invariants of X. These calculations can be prohibitively expensive if the number of cells in the approximating complex S is large. Unfortunately, most existing simplicial approximation algorithms either give too many cells, or involve calculations which are tractable or valid only in low dimensional Euclidean geometry. In this paper we introduce the combinatorial Delaunay triangulation, a simplicial complex construction which can be efficiently computed in arbitrary metric spaces, and which gives reliable topological approximations using comparatively few cells.

19th International Meshing Roundtable (IMR '10), 2010

We consider here the problem of representing non-manifold shapes discretized as d-dimensional simplicial Euclidean complexes. To this aim, we propose a dimension-independent data structure for simplicial complexes, called the Incidence Simplicial (IS) data structure, which is scalable to manifold complexes, and supports efficient navigation and topological modifications. The IS data structure has the same expressive power and exhibits performances in the query and update operations as the incidence graph, a widely-used representation for general cell complexes, but it is much more compact. Here, we describe the IS data structure and we evaluate its storage cost. Moreover, we present efficient algorithms for navigating and for generating a simplicial complex described as an IS data structure. We compare the IS data structure with the incidence graph and with dimension-specific representations for simplicial complexes.

2022

The use of topological descriptors in modern machine learning applications, such as Persistence Diagrams (PDs) arising from Topological Data Analysis (TDA), has shown great potential in various domains. However, their practical use in applications is often hindered by two major limitations: the computational complexity required to compute such descriptors exactly, and their sensitivity to even low-level proportions of outliers. In this work, we propose to bypass these two burdens in a data-driven setting by entrusting the estimation of (vectorization of) PDs built on top of point clouds to a neural network architecture that we call RipsNet. Once trained on a given data set, RipsNet can estimate topological descriptors on test data very efficiently with generalization capacity. Furthermore, we prove that RipsNet is robust to input perturbations in terms of the 1-Wasserstein distance, a major improvement over the standard computation of PDs that only enjoys Hausdorff stability, yieldi...

2016

This paper presents a set of tools to compute topological information of simplicial complexes, tools that are applicable to extract topological information from digital pictures. A simplicial complex is encoded in a (non-unique) algebraic-topological format called AM-model. An AM-model for a given object K is determined by a concrete chain homotopy and it provides, in particular, integer (co)homology generators of K and representative (co)cycles of these generators. An algorithm for computing an AM-model and the cohomological invariant HB1 (derived from the rank of the cohomology ring) with integer coefficients for a finite simplicial complex in any dimension is designed here, extending the work done in [9] in which the ground ring was a field. A concept of generators which are "nicely" representative is also presented. Moreover, we extend the definition of AM-models to 3D binary digital images and we design algorithms to update the AM-model information after voxel set operations (union, intersection, difference and inverse).

Discrete Applied Mathematics, 2009

This paper presents a set of tools to compute topological information of simplicial complexes, tools that are applicable to extract topological information from digital pictures. A simplicial complex is encoded in a (non-unique) algebraic-topological format called AM-model. An AM-model for a given object K is determined by a concrete chain homotopy and it provides, in particular, integer (co)homology generators of K and representative (co)cycles of these generators. An algorithm for computing an AM-model and the cohomological invariant HB1 (derived from the rank of the cohomology ring) with integer coefficients for a finite simplicial complex in any dimension is designed here, extending the work done in in which the ground ring was a field. A concept of generators which are "nicely" representative is also presented. Moreover, we extend the definition of AM-models to 3D binary digital images and we design algorithms to update the AM-model information after voxel set operations (union, intersection, difference and inverse).

2009

Recently several types of complexes have been proposed for topological analysis of data lying on a manifold in a high dimensional space. The effectiveness of the method in practice surely depends on the computational costs of constructing these complexes. The complexes such as restricted Delaunay, alpha complex,Čech and witness complex are difficult to compute in high dimensions. As an alternative, Rips complex, a well known structure in algebraic topology, has been proposed for computing homological information. While their computations are easy, their size tends to be large. We propose a Rips-like complex called geodesic complex which has smaller size than the standard Rips complex. The gain in size results from the fact that a geodesic complex is built by approximating intrinsic distances on the embedded manifold whereas a Rips complex is built with extrinsic distances in the embedding space. In the course of the development, we connect among various existing results which may find further use in topological analysis of data.

Lecture Notes in Computer Science, 2012

Let K be a simplicial complex and g the rank of its p-th homology group H p (K) defined with Z 2 coefficients. We show that we can compute a basis H of H p (K) and annotate each p-simplex of K with a binary vector of length g with the following property: the annotations, summed over all p-simplices in any p-cycle z, provide the coordinate vector of the homology class [z] in the basis H. The basis and the annotations for all simplices can be computed in O(n ω) time, where n is the size of K and ω < 2.376 is a quantity so that two n × n matrices can be multiplied in O(n ω) time. The pre-computation of annotations permits answering queries about the independence or the triviality of p-cycles efficiently. Using annotations of edges in 2-complexes, we derive better algorithms for computing optimal basis and optimal homologous cycles in 1-dimensional homology. Specifically, for computing an optimal basis of H 1 (K), we improve the time complexity known for the problem from O(n 4) to O(n ω +n 2 g ω−1). Here n denotes the size of the 2-skeleton of K and g the rank of H 1 (K). Computing an optimal cycle homologous to a given 1-cycle is NP-hard even for surfaces and an algorithm taking O(2 O(g) n log n) time is known for surfaces. We extend this algorithm to work with arbitrary 2-complexes in O(n ω + 2 O(g) n 2 log n) time using annotations.

Proceedings of the Web Conference 2021, 2021

A simplicial complex is a generalization of a graph: a collection of =-ary relationships (instead of binary as the edges of a graph), named simplices. In this paper, we develop a new tool to study the structure of simplicial complexes: we generalize the graph notion of truss decomposition to complexes, and show that this more powerful representation gives rise to dierent properties compared to the graph-based one. This power, however, comes with important computational challenges derived from the combinatorial explosion caused by the downward closure property of complexes. Drawing upon ideas from itemset mining and similarity search, we design a memory-aware algorithm, dubbed STD, which is able to eciently compute the truss decomposition of a simplicial complex. STD adapts its behavior to the amount of available memory by storing intermediate data in a compact way. We then devise a variant that computes directly the = simplices of maximum trussness. By applying STD to several datasets, we prove its scalability, and provide an analysis of their structure. Finally, we show that the truss decomposition can be seen as a ltration, and as such it can be used to study the persistent homology of a dataset, a method for computing topological features at dierent spatial resolutions, prominent in Topological Data Analysis.