13 research outputs found
The Morse theory of \v{C}ech and Delaunay complexes
Given a finite set of points in and a radius parameter, we
study the \v{C}ech, Delaunay-\v{C}ech, Delaunay (or Alpha), and Wrap complexes
in the light of generalized discrete Morse theory. Establishing the \v{C}ech
and Delaunay complexes as sublevel sets of generalized discrete Morse
functions, we prove that the four complexes are simple-homotopy equivalent by a
sequence of simplicial collapses, which are explicitly described by a single
discrete gradient field.Comment: 21 pages, 2 figures, improved expositio
LIPIcs
Given a locally finite X â âd and a radius r â„ 0, the k-fold cover of X and r consists of all points in âd that have k or more points of X within distance r. We consider two filtrations - one in scale obtained by fixing k and increasing r, and the other in depth obtained by fixing r and decreasing k - and we compute the persistence diagrams of both. While standard methods suffice for the filtration in scale, we need novel geometric and topological concepts for the filtration in depth. In particular, we introduce a rhomboid tiling in âd+1 whose horizontal integer slices are the order-k Delaunay mosaics of X, and construct a zigzag module from Delaunay mosaics that is isomorphic to the persistence module of the multi-covers
When Convexity Helps Collapsing Complexes
This paper illustrates how convexity hypotheses help collapsing simplicial complexes. We first consider a collection of compact convex sets and show that the nerve of the collection is collapsible whenever the union of sets in the collection is convex. We apply this result to prove that the Delaunay complex of a finite point set is collapsible. We then consider a convex domain defined as the convex hull of a finite point set. We show that if the point set samples sufficiently densely the domain, then both the Cech complex and the Rips complex of the point set are collapsible for a well-chosen scale parameter. A key ingredient in our proofs consists in building a filtration by sweeping space with a growing sphere whose center has been fixed and studying events occurring through the filtration. Since the filtration mimics the sublevel sets of a Morse function with a single critical point, we anticipate this work to lay the foundations for a non-smooth, discrete Morse Theory
Topological Data Analysis
International audienceIt has been observed since a long time that data are often carrying interesting topological and geometric structures. Characterizing such structures and providing efficient tools to infer and exploit them is a challenging problem that asks for new mathematics and that is motivated by a real need from applications. This paper is an introduction to Topological Data Analysis (), a new field that emerged during the last two decades with the objective of understanding and exploiting the topological structure of modern and complex data. The paper surveys some important mathematical and algorithmic developments in as well as software solutions that are currently used to address various applied and industrial problems
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Topological and geometric inference of data
The overarching problem under consideration is to determine the structure
of the subspace on which a distribution is supported, given
only a finite noisy sample thereof. The special case in
which the subspace is an embedded manifold is given particular
attention owing to its conceptual elegance, and asymptotic bounds are
obtained on the admissible level of noise such that the
manifold can be recovered up to homotopy equivalence.
Attention is turned on how to accomplish this in practice.
Following ideas from topological data analysis, simplicial complexes are used
as discrete analogues of spaces suitable for computation. By utilising
the prior assumption that the data lie on a manifold, topologically
inspired techniques are proposed for refining the simplicial complex
to better approximate this manifold. This is applied to the
problem of nonlinear dimensionality reduction and found to improve accuracy
of reconstructing several synthetic and real-world datasets.
The second chapter focuses on extending this work to the
case where the ambient space is non-Euclidean. The interfaces between
topological data analysis, functional data analysis, and shape analysis
are thoroughly explored. Lipschitz bounds are proved which relate several
metrics on the space of positive semidefinite matrices; they are then
interpreted in the context of topological data analysis. This is
applied to diffusion tensor imaging and phonology.
The final chapter explores the case where the points are
non-uniformly distributed over the embedded subspace. In particular, a method
is proposed to overcome the shortcomings of witness complex construction
when there are large deviations in the density. The theory
of multidimensional persistence is leveraged to provide a succinct setting
in which the structure of the data can be interpreted
as a generalised stratified space.EPSR
A Unified View on the Functorial Nerve Theorem and its Variations
The nerve theorem is a basic result of algebraic topology that plays a
central role in computational and applied aspects of the subject. In applied
topology, one often needs a nerve theorem that is functorial in an appropriate
sense, and furthermore one often needs a nerve theorem for closed covers, as
well as for open covers. While the techniques for proving such functorial nerve
theorems have long been available, there is unfortunately no general-purpose,
explicit treatment of this topic in the literature. We address this by proving
a variety of functorial nerve theorems. First, we show how one can use
relatively elementary techniques to prove nerve theorems for covers by closed
convex sets in Euclidean space, and for covers of a simplicial complex by
subcomplexes. Then, we prove a more general, "unified" nerve theorem that
recovers both of these, using standard techniques from abstract homotopy
theory.Comment: 53 pages. Updated exposition and added Appendix D. Comments welcom
Topological data analysis in information space
Various kinds of data are routinely represented as discrete probability distributions. Examples include text documents summarized by histograms of word occurrences and images represented as histograms of oriented gradients. Viewing a discrete probability distribution as a point in the standard simplex of the appropriate dimension, we can understand collections of such objects in geometric and topological terms. Importantly, instead of using the standard Euclidean distance, we look into dissimilarity measures with information-theoretic justification, and we develop the theory needed for applying topological data analysis in this setting. In doing so, we emphasize constructions that enable the usage of existing computational topology software in this context
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
Simplicial Data Analysis: theory, practice, and algorithms
Simplicial complexes store in discrete form key information on a topological space, and have been used in mathematics to introduce combinatorial and discrete tools in geometry and topology. They represent a topological space as a collection of âsimple elementsâ (such as vertices, edges, triangles, tetrahedra, and more general simplices) that are glued to each other in a structured manner. In the last 40 years, they have been a basic tool in computer visualization for storing and classifying different shapes of 3d images, then in the early 2000s these techniques were success- fully applied to more general data, not necessarily sampled from a metric space.
The use of techniques borrowed from algebraic topology has been very successfull in analysing data from various fields: genomics, sensor analysis, brain connectomics, fMRI data, trade net- works, and new fields of application are being tested every day. Regrettably, topological data analysis has been used mainly as a qualitative method, the problem being the lack of proper tools to perform effective statistical analysis.
Coming from well established techniques in random graph theory, the first models for random simplicial complexes have been introduced in recent years, none of which though can be used effectively in a quantitative analysis of data. We introduce a model that can be successfully used as a null model for simplicial complexes as it fixes the size distribution of facets.
Another challenge is to successfully identify a simplicial complex which can correctly encode the topological space from which the initial data set is sampled. The most common solution is to build nesting simplicial complexes, and study the evolution of their features. A recent study uncovered that the problem can reside in making wrong assumption on the space of data. We propose a categorical reasoning which enlightens the cause leading to these misconceptions. We introduce a new category for weighted graphs and study its relation to other common categories when the weights are chosen in a poset.
The construction of the appropriate simplicial complex is not the only obstacle one faces when applying topological methods to real data. Available algorithms for homological features extraction have a memory and time complexity which scales exponentially on the number of simplices, making these techniques not suitable for the analysis of âbig dataâ. We propose a quantum algorithm which is able to track in logarithmic time the evolution of a quantum version of well known homological features along a filtration of simplicial complexes
Applications de lâhomologie persistante pour la reconnaissance des formes
Lâhomologie persistante est un outil fondamental dans la topologie computationnelle. Cette mĂ©thode est utilisĂ©e pour reconnaĂźtre et comparer les formes. Dans ce travail nous Ă©tudions dâabord lâhomologie persistante dans le cas unidimensionnel dâordre 0 quâon appelle aussi fonction de taille. Nous prĂ©sentons une dĂ©monstration du fait que toute fonction de taille peut ĂȘtre reprĂ©sentĂ©e comme un ensemble de points et de lignes dans le plan rĂ©el, avec des multiplicitĂ©s. Cela permet une approche algĂ©brique aux fonctions de taille et la construction de nouvelles pseudo distances entre les fonctions de taille pour comparer les formes. Nous calculons ensuite lâhomologie persistante unidimensionnelle dâordre n avec diffĂ©rentes mĂ©thodes de filtration de lâespace correspondant Ă lâhistoire dâun complexe croissant. Nous classons un changement topologique qui se produit pendant la croissance soit comme une caractĂ©ristique ou un bruit, en fonction de sa durĂ©e de vie ou de sa persistance dans la filtration. Une prĂ©sentation avec des codes barres affiche alors la persistance de ces invariants. Lâhomologie persistante multidimensionnelle nous permet de soutirer plus dâinformations sur les formes en utilisant la fonction de filtration avec des valeurs dans [nombre rĂ©el]k. Pour fournir un descripteur de forme concis et complet dans le cas multidimensionnel nous rĂ©duisons le calcul de lâhomologie persistante multidimensionnelle au calcul de lâhomologie persistante ordinaire pour une famille paramĂ©trĂ©e de fonctions Ă valeur dans [nombre rĂ©el]