Topological inference via meshing

Abstract

We apply ideas from mesh generation to improve the time and space complexities of computing the full persistent homological information associated with a point cloud P in Euclidean space R d . Classical approaches rely on theČech, Rips, α-complex, or witness complex filtrations of P , whose complexities scale up very badly with d. For instance, the α-complex filtration incurs the n Ω(d) size of the Delaunay triangulation, where n is the size of P . The common alternative is to truncate the filtrations when the sizes of the complexes become prohibitive, possibly before discovering the most relevant topological features. In this paper we propose a new collection of filtrations, based on the Delaunay triangulation of a carefully-chosen superset of P , whose sizes are reduced to 2 O(d 2 ) n. A nice property of these filtrations is to be interleaved multiplicatively with the family of offsets of P , so that the persistence diagram of P can be approximated in 23 time in theory, with a near-linear observed running time in practice (ignoring the constant factors depending exponentially on d). Thus, our approach remains tractable in medium dimensions, say 4 to 10.Key-words: Topological persistence, Delaunay triangulation, offsets, sparse Voronoi refinement. * bhudson@tti-c.edu † glmiller@cs.cmu.edu ‡ steve.oudot@inria.fr § dsheehy@cs.cmu.edu Inférence topologique par maillageRésumé : Nous appliquons des idées issues de la litérature sur la génération de maillages afin d'eméliorer la complexité du calcul de l'information topologique persistante associéesà un nuage de poins P dans l'espace euclidien R d . Les méthodes classiques reposent sur l'utilisation de filtrations telles que celle deČech, celle de Rips-Vietoris, celle de l'α-complex ou celle du witness complex, dont les complexités se comportent très mal lorsque la dimension d augmente. Par exemple, la filtration de l'α-complex a la même taille que la triangulation de Delaunay de P , qui est de l'ordre de n Ω(d( dans le pire cas, où n est la taille de P . La solution communément adoptée consisteà tronquer les filtrations avant que leur coût ne devienne prohibitif, mais bien sûr peut-êtreégalement avant que les données topologiques les plus pertinentes n'aientété saisies. Dans cet article nous proposons une nouvelle famille de filtrations, basée sur la triangulation de Delaunay d'un sur-ensemble fini M de P , dont la taille est réduiteà 2 O(d 2 ) n. Une propriété intéressante de ces filtrations est d'être entrelacées multiplicativement avec la famille des offsets de P , si bien que le diagramme de persistance de P peutêtre approché en temps 2en théorie, avec un comportement quasi-linéaire en pratique (en laissant de côté les facteurs dépendant exponentiellement en la dimension d). Ainsi, notre approche demeure praticable en dimensions moyennes, disons entre 4 et 10.

Zigzag Zoology: Rips Zigzags for Homology Inference

“…A different approach to the problem of building sparse filtrations for offsets of point clouds in Euclidean space was presented by Hudson et al [22]. They used ideas from Delaunay refinement mesh generation to build linear-size filtrations that provide provably good approximations to the persistence diagram of the offsets.…”

“…As a proof of concept, we ran our code on the so-called Clifford data set from [22], which was obtained by evenly spacing 2,000 points along the line l : y = 20x mod 2π in the second flat torus (R mod 2π) 2 , then mapping the points onto the Clifford torus in R 4 via the embedding f : (u, v) → (cos u, sin u, cos v, sin v). This data set admits three nontrivial candidate underlying spaces: at small scales, the image of l through f , which is a closed helicoidal curve on the torus; at larger scales, the torus itself; at even larger scales, the 3-sphere of radius √ 2 on which the torus is sitting.…”

“…This delicate 3-cycle certainly exists in the barcode of the standard Rips filtration; however, it cannot be observed in practice. As mentioned in [22], the union of balls of radius α around the data points covers the entire sphere only for α ≥ 4 − 2 √ 2, so the corresponding 3-homology cycle can appear in the Rips filtration only at a parameter value α ≥…”

A fast algorithm for well-spaced points and approximate delaunay graphs

“…Delaunay meshes are used extensively in graphics and scientific computing, and more recently, they have been applied in higher dimensions for data analysis [3,10]. However, there are serious di culties in constructing meshes in more than three dimensions.…”

Persistent Homology and Nested Dissection

“…This includes primarily filtrations constructed on grids or quality meshes. Hudson et al [HMOS10] show that the persistent homology of the Euclidean distance to a point cloud can be approximated on the type of mesh used in finite element analysis. Sheehy [She11] extends this result to a large class of Lipschitz functions.…”