The Vietoris-Rips filtration is a versatile tool in topological data analysis. It is a sequence of simplicial complexes built on a metric space to add topological structure to an otherwise disconnected set of points. It is widely used because it encodes useful information about the topology of the underlying metric space. This information is often extracted from its so-called persistence diagram. Unfortunately, this filtration is often too large to construct in full. We show how to construct an O(n)-size filtered simplicial complex on an n-point metric space such that its persistence diagram is a good approximation to that of the Vietoris-Rips filtration. This new filtration can be constructed in O(n log n) time. The constant factors in both the size and the running time depend only on the doubling dimension of the metric space and the desired tightness of the approximation. For the first time, this makes it computationally tractable to approximate the persistence diagram of the Vietoris-Rips filtration across all scales for large data sets.We describe two different sparse filtrations. The first is a zigzag filtration that removes points as the scale increases. The second is a (non-zigzag) filtration that yields the same persistence diagram. Both methods are based on a hierarchical net-tree and yield the same guarantees.

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We present the Iterated-Tverberg algorithm, the first deterministic algorithm for computing an approximate centerpoint of a set S ∈ R d with running time sub-exponential in d. The algorithm is a derandomization of the Iterated-Radon algorithm of Clarkson et al and is guaranteed to terminate with an O(1/d 2)-center. Moreover, it returns a polynomial-time checkable proof of the approximation guarantee, despite the coNP-Completenes of testing centerpoints in general. We also explore the use of higher order Tverberg partitions to improve the runtime of the deterministic algorithm and improve the approximation guarantee for the randomized algorithm. In particular, we show how to improve the O(1/d 2)-center of the Iterated-Radon algorithm to O(1/d r r−1) for a cost of O((rd) d) in time for any integer r.

Given n points P in a Euclidean space, the Johnson-Linden-strauss lemma guarantees that the distances between pairs of points is preserved up to a small constant factor with high probability by random projection into O(log n) dimensions. In this paper, we show that the persistent homology of the distance function to P is also preserved up to a comparable constant factor. One could never hope to preserve the distance function to P pointwise, but we show that it is preserved sufficiently at the critical points of the distance function to guarantee similar persistent homology. We prove these results in the more general setting of weighted kth nearest neighbor distances, for which k = 1 and all weights equal to zero gives the usual distance to P .

The Vietoris-Rips filtration is a versatile tool in topological data analysis. It is a sequence of simplicial complexes built on a metric space to add topological structure to an otherwise disconnected set of points. It is widely used because it encodes useful information about the topology of the underlying metric space. This information is often extracted from its so-called persistence diagram. Unfortunately, this filtration is often too large to construct in full. We show how to construct an O(n)-size filtered simplicial complex on an n-point metric space such that its persistence diagram is a good approximation to that of the Vietoris-Rips filtration. This new filtration can be constructed in O(n log n) time. The constant factors in both the size and the running time depend only on the doubling dimension of the metric space and the desired tightness of the approximation. For the first time, this makes it computationally tractable to approximate the persistence diagram of the Vietoris-Rips filtration across all scales for large data sets. We describe two different sparse filtrations. The first is a zigzag filtration that removes points as the scale increases. The second is a (non-zigzag) filtration that yields the same persistence diagram. Both methods are based on a hierarchical net-tree and yield the same guarantees.

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.

A new paradigm for point cloud data analysis has emerged recently, where point clouds are no longer treated as mere compact sets but rather as empirical measures. A notion of distance to such measures has been defined and shown to be stable with respect to perturbations of the measure. This distance can easily be computed pointwise in the case of a point cloud, but its sublevel-sets, which carry the geometric information about the measure, remain hard to compute or approximate. This makes it challenging to adapt many powerful techniques based on the Euclidean distance to a point cloud to the more general setting of the distance to a measure on a metric space.We propose an efficient and reliable scheme to approximate the topological structure of the family of sublevel-sets of the distance to a measure. We obtain an algorithm for approximating the persistent homology of the distance to an empirical measure that works in arbitrary metric spaces. Precise quality and complexity guarantees are given with a discussion on the behavior of our approach in practice.

Many surfactant-based formulations are utilised in industry as they produce desirable visco-elastic properties at low-concentrations. These properties are due to the presence of worm-like micelles (WLM) and, as a result, understanding the processes that