Section: New Results
Algorithmic aspects of topological and geometric data analysis
DTM-based filtrations
Participants : Frédéric Chazal, Marc Glisse, Raphaël Tinarrage.
In collaboration with H. Anai, Y. Ike, H. Inakoshi and Y. Umeda of Fujitsu.
Despite strong stability properties, the persistent homology of filtrations classically used in Topological Data Analysis, such as, e.g. the Čech or Vietoris-Rips filtrations, are very sensitive to the presence of outliers in the data from which they are computed. In this paper [33], we introduce and study a new family of filtrations, the DTM-filtrations, built on top of point clouds in the Euclidean space which are more robust to noise and outliers. The approach adopted in this work relies on the notion of distance-to-measure functions, and extends some previous work on the approximation of such functions.
Persistent Homology with Dimensionality Reduction: -Distance vs Gaussian Kernels
Participants : Shreya Arya, Jean-Daniel Boissonnat, Kunal Dutta.
We investigate the effectiveness of dimensionality reduction for computing the persistent homology for both
Computing Persistent Homology of Flag Complexes via Strong Collapses
Participants : Jean-Daniel Boissonnat, Siddharth Pritam.
In collaboration with Divyansh Pareek (Indian Institute of Technology Bombay, India)
We introduce a fast and memory efficient approach to compute the persistent homology (PH) of a sequence of simplicial complexes. The basic idea is to simplify the complexes of the input sequence by using strong collapses, as introduced by J. Barmak and E. Miniam [DCG (2012)], and to compute the PH of an induced sequence of reduced simplicial complexes that has the same PH as the initial one. Our approach has several salient features that distinguishes it from previous work. It is not limited to filtrations (i.e. sequences of nested simplicial subcomplexes) but works for other types of sequences like towers and zigzags. To strong collapse a simplicial complex, we only need to store the maximal simplices of the complex, not the full set of all its simplices, which saves a lot of space and time. Moreover, the complexes in the sequence can be strong collapsed independently and in parallel. Finally, we can compromize between precision and time by choosing the number of simplicial complexes of the sequence we strong collapse. As a result and as demonstrated by numerous experiments on publicly available data sets, our approach is extremely fast and memory efficient in practice [27].
Strong Collapse for Persistence
Participants : Jean-Daniel Boissonnat, Siddharth Pritam.
In this paper, we build on the initial success of and show that further decisive progress can be obtained if one restricts the family of simplicial complexes to flag complexes. Flag complexes are fully characterized by their graph (or 1-skeleton), the other faces being obtained by computing the cliques of the graph. Hence, a flag complex can be represented by its graph, which is a very compact representation. Flag complexes are very popular and, in particular, Vietoris-Rips complexes are by far the most widely simplicial complexes used in Topological Data Analysis. It has been shown in that the persistent homology of Vietoris-Rips filtrations can be computed very efficiently using strong collapses. However, most of the time was devoted to computing the maximal cliques of the complex prior to their strong collapse.
In this paper [37], we observe that the reduced complex obtained by strong collapsing a flag complex is itself a flag complex. Moreover, this reduced complex can be computed using only the 1-skeleton (or graph) of the complex, not the set of its maximal cliques. Finally, we show how to compute the equivalent filtration of the sequence of reduced flag simplicial complexes using again only 1-skeletons. x On the theory side, we show that strong collapses of flag complexes can be computed in time
Triangulating submanifolds: An elementary and quantified version of Whitney's method
Participants : Jean-Daniel Boissonnat, Siargey Kachanovich, Mathijs Wintraecken.
We quantize Whitney's construction to prove the existence of a triangulation for any
Randomized incremental construction of Delaunay triangulations of nice point sets
Participants : Jean-Daniel Boissonnat, Kunal Dutta, Marc Glisse.
In collaboration with Olivier Devillers (Inria, CNRS, Loria, Université de Lorraine).
Randomized incremental construction (RIC) is one of the most important paradigms for building geometric data structures. Clarkson and Shor developed a general theory that led to numerous algorithms that are both simple and efficient in theory and in practice.
Randomized incremental constructions are most of the time space and time optimal in the worst-case, as exemplified by the construction of convex hulls, Delaunay triangulations and arrangements of line segments.
However, the worst-case scenario occurs rarely in practice and we
would like to understand how RIC behaves when the input is nice in
the sense that the associated output is significantly smaller than
in the worst-case. For example,
it is known that the Delaunay triangulations of nicely distributed points in
Along the way, we prove a probabilistic lemma for sampling without replacement, which may be of independent interest.
Approximate Polytope Membership Queries
Participant : Guilherme Da Fonseca.
In collaboration with Sunil Arya (Hong Kong University of Science and Technology) and David Mount (University of Maryland).
In the polytope membership problem, a convex polytope
Approximate Convex Intersection Detection with Applications to Width and Minkowski Sums
Participant : Guilherme Da Fonseca.
In collaboration with Sunil Arya (Hong Kong University of Science and Technology) and David Mount (University of Maryland).
Approximation problems involving a single convex body in
Approximating the Spectrum of a Graph
Participant : David Cohen-Steiner.
In collaboration with Weihao Kong (Stanford University), Christian Sohler (TU Dortmund) and Gregory Valiant (Stanford University).
The spectrum of a network or graph
Spectral Properties of Radial Kernels and Clustering in High Dimensions
Participants : David Cohen-Steiner, Alba Chiara de Vitis.
In this paper [40], we study the spectrum and the eigenvectors of radial kernels for mixtures of distributions in
Exact computation of the matching distance on 2-parameter persistence modules
Participant : Steve Oudot.
In collaboration with Michael Kerber (T.U. Graz) and Michael Lesnick (SUNY).
The matching distance is a pseudometric on multi-parameter persistence modules, defined in terms of the weighted bottleneck distance on the restriction of the modules to affine lines. It is known that this distance is stable in a reasonable sense, and can be efficiently approximated, which makes it a promising tool for practical applications. In [44] we show that in the 2-parameter setting, the matching distance can be computed exactly in polynomial time. Our approach subdivides the space of affine lines into regions, via a line arrangement. In each region, the matching distance restricts to a simple analytic function, whose maximum is easily computed. As a byproduct, our analysis establishes that the matching distance is a rational number, if the bigrades of the input modules are rational.
A Comparison Framework for Interleaved Persistence Modules
Participant : Miroslav Kramár.
In collaboration with Rachel Levanger (UPenn), Shaun Harker and Konstantin Mischaikow (Rutgers).
In [43], we present a generalization of the induced matching theorem of [1] and use it to prove a generalization of the algebraic stability theorem for R-indexed pointwise finite-dimensional persistence modules. Via numerous examples, we show how the generalized algebraic stability theorem enables the computation of rigorous error bounds in the space of persistence diagrams that go beyond the typical formulation in terms of bottleneck (or log bottleneck) distance.
Discrete Morse Theory for Computing Zigzag Persistence
Participant : Clément Maria.
In collaboration with Hannah Schreiber (Graz University of Technology, Austria)
We introduce a framework to simplify zigzag filtrations of general complexes using discrete Morse theory, in order to accelerate the computation of zigzag persistence. Zigzag persistence is a powerful algebraic generalization of persistent homology. However, its computation is much slower in practice, and the usual optimization techniques cannot be used to compute it. Our approach is different in that it preprocesses the filtration before computation. Using discrete Morse theory, we get a much smaller zigzag filtration with same persistence. The new filtration contains general complexes. We introduce new update procedures to modify on the fly the algebraic data (the zigzag persistence matrix) under the new combinatorial changes induced by the Morse reduction. Our approach is significantly faster in practice [45].