Section: New Results

Algorithmic aspects of topological and geometric data analysis

Variance Minimizing Transport Plans for Inter-surface Mapping

Participant : David Cohen-Steiner.

In collaboration with Manish Mandad, Leik Kobbelt (RWTH Aachen), Pierre Alliez (Inria), and Mathieu Desbrun (Caltech).

We introduce an effcient computational method for generating dense and low distortion maps between two arbitrary surfaces of same genus. Instead of relying on semantic correspondences or surface parameterization, we directly optimize a variance-minimizing transport plan between two input surfaces that defines an as-conformal-as-possible inter-surface map satisfying a user-prescribed bound on area distortion. The transport plan is computed via two alternating convex optimizations, and is shown to minimize a generalized Dirichlet energy of both the map and its inverse. Computational efficiency is achieved through a coarse-tone approach in diffusion geometry, with Sinkhorn iterations modified to enforce bounded area distortion. The resulting inter-surface mapping algorithm applies to arbitrary shapes robustly, with little to no user interaction.

Approximating the spectrum of a graph

Participant : David Cohen-Steiner.

In collaboration with Weihao Kong, Gregory Valiant (Stanford), and Christian Sohler (TU Dortmund).

The spectrum of a network or graph $G=(V,E)$ with adjacency matrix $A$ consists of the eigenvalues of the normalized Laplacian $L=I-D^{-1/2}A{D}^{-1/2}$. This set of eigenvalues encapsulates many aspects of the structure of the graph, including the extent to which the graph posses community structures at multiple scales. We study the problem of approximating the spectrum $\lambda =({\lambda }_1,\dots ,{\lambda }_{\left|V\right|})$, $0\le \lambda 1,\le \dots ,\le \lambda \left|V\right|\le 2$ of $G$ in the regime where the graph is too large to explicitly calculate the spectrum. We present a sublinear time algorithm that, given the ability to query a random node in the graph and select a random neighbor of a given node, computes a succinct representation of an approximation $\tilde{\lambda }$ such that $\parallel \tilde{\lambda }{-\lambda \parallel }_1\le \epsilon \left|V\right|$. Our algorithm has query complexity and running time $exp\left(O\right(1/\epsilon \left)\right)$ , independent of the size of the graph, $\left|V\right|$. We demonstrate the practical viability of our algorithm on 15 different real-world graphs from the Stanford Large Network Dataset Collection, including social networks, academic collaboration graphs, and road networks. For the smallest of these graphs, we are able to validate the accuracy of our algorithm by explicitly calculating the true spectrum; for the larger graphs, such a calculation is computationally prohibitive. In addition we study the implications of our algorithm to property testing in the bounded degree graph model.

Anisotropic triangulations via discrete Riemannian Voronoi diagrams

Participants : Jean-Daniel Boissonnat, Mathijs Wintraecken.

In collaboration with mael Rouxel-Labbé (GeometryFactory).

The construction of anisotropic triangulations is desirable for various applications, such as the numerical solving of partial differential equations and the representation of surfaces in graphics. To solve this notoriously difficult problem in a practical way, we introduce the discrete Riemannian Voronoi diagram, a discrete structure that approximates the Riemannian Voronoi diagram. This structure has been implemented and was shown to lead to good triangulations in ${ℝ}^2$ and on surfaces embedded in ${ℝ}^3$ as detailed in our experimental companion paper.

In [23], [32], [34], we study theoretical aspects of our structure. Given a finite set of points $𝒫$ in a domain $\Omega$ equipped with a Riemannian metric, we compare the discrete Riemannian Voronoi diagram of $𝒫$ to its Riemannian Voronoi diagram. Both diagrams have dual structures called the discrete Riemannian Delaunay and the Riemannian Delaunay complex. We provide conditions that guarantee that these dual structures are identical. It then follows from previous results that the discrete Riemannian Delaunay complex can be embedded in $\Omega$ under sufficient conditions, leading to an anisotropic triangulation with curved simplices. Furthermore, we show that, under similar conditions, the simplices of this triangulation can be straightened.

Only distances are required to reconstruct submanifolds

Participants : Jean-Daniel Boissonnat, Ramsay Dyer, Steve Oudot.

In collaboration with Arijit Ghosh (Indian Statistical Institute).

In [14], we give the first algorithm that outputs a faithful reconstruction of a submanifold of Euclidean space without maintaining or even constructing complicated data structures such as Voronoi diagrams or Delaunay complexes. Our algorithm uses the witness complex and relies on the stability of power protection, a notion introduced in this paper. The complexity of the algorithm depends exponentially on the intrinsic dimension of the manifold, rather than the dimension of ambient space, and linearly on the dimension of the ambient space. Another interesting feature of this work is that no explicit coordinates of the points in the point sample is needed. The algorithm only needs the distance matrix as input, i.e., only distance between points in the point sample as input.

An obstruction to Delaunay triangulations in Riemannian manifolds

Participants : Jean-Daniel Boissonnat, Ramsay Dyer.

In collaboration with Arijit Ghosh (Indian Statistical Institute) and Nikolay Martynchuk (University of Groningen).

Delaunay has shown that the Delaunay complex of a finite set of points $P$ of Euclidean space ${ℝ}^m$ triangulates the convex hull of $P$, provided that $P$ satisfies a mild genericity property. Voronoi diagrams and Delaunay complexes can be defined for arbitrary Riemannian manifolds. However, Delaunay's genericity assumption no longer guarantees that the Delaunay complex will yield a triangulation; stronger assumptions on $P$ are required. A natural one is to assume that $P$ is sufficiently dense. Although results in this direction have been claimed, we show that sample density alone is insufficient to ensure that the Delaunay complex triangulates a manifold of dimension greater than 2 [13].

Local criteria for triangulation of manifolds

Participants : Jean-Daniel Boissonnat, Ramsay Dyer, Mathijs Wintraecken.

In collaboration with Arijit Ghosh (Indian Statistical Institute).

We present criteria for establishing a triangulation of a manifold [40]. Given a manifold $M$, a simplicial complex $𝒜$, and a map $H$ from the underlying space of $𝒜$ to $M$, our criteria are presented in local coordinate charts for $M$, and ensure that $H$ is a homeomorphism. These criteria do not require a differentiable structure, or even an explicit metric on $M$. No Delaunay property of $𝒜$ is assumed. The result provides a triangulation guarantee for algorithms that construct a simplicial complex by working in local coordinate patches. Because the criteria are easily checked algorithmically, they are expected to be of general use.

Triangulating stratified manifolds I: a reach comparison theorem

Participants : Jean-Daniel Boissonnat, Mathijs Wintraecken.

In [42], we define the reach for submanifolds of Riemannian manifolds, in a way that is similar to the Euclidean case. Given a $d$-dimensional submanifold $𝒮$ of a smooth Riemannian manifold $𝕄$ and a point $p\in 𝕄$ that is not too far from $𝒮$ we want to give bounds on local feature size of ${exp}_p^{-1}\left(𝒮\right)$. Here ${exp}_p^{-1}$ is the inverse exponential map, a canonical map from the manifold to the tangent space. Bounds on the local feature size of ${exp}_p^{-1}\left(𝒮\right)$ can be reduced to giving bounds on the reach of ${exp}_p^{-1}\left(ℬ\right)$, where $ℬ$ is a geodesic ball, centred at $c$ with radius equal to the reach of $𝒮$. Equivalently we can give bounds on the reach of ${exp}_p^{-1}\circ {exp}_c\left(B_c\right)$, where now $B_c$ is a ball in the tangent space $T_c𝕄$, with the same radius. To establish bounds on the reach of ${exp}_p^{-1}\circ {exp}_c\left(B_c\right)$ we use bounds on the metric and on its derivative in Riemann normal coordinates.

This result is a first step towards answering the important question of how to triangulate stratified manifolds.

The reach, metric distortion, geodesic convexity and the variation of tangent spaces

Participants : Jean-Daniel Boissonnat, Mathijs Wintraecken.

In collaboration with André Lieutier (Dassault Système).

In [41], we discuss three results. The first two concern general sets of positive reach: We first characterize the reach by means of a bound on the metric distortion between the distance in the ambient Euclidean space and the set of positive reach. Secondly, we prove that the intersection of a ball with radius less than the reach with the set is geodesically convex, meaning that the shortest path between any two points in the intersection lies itself in the intersection. For our third result we focus on manifolds with positive reach and give a bound on the angle between tangent spaces at two different points in terms of the distance between the points and the reach.

Delaunay triangulation of a random sample of a good sample has linear size

Participants : Jean-Daniel Boissonnat, Kunal Dutta, Marc Glisse.

In collaboration with Olivier Devillers (Inria Nancy Grand Est).

The randomized incremental construction (RIC) for building geometric data structures has been analyzed extensively, from the point of view of worst-case distributions. In many practical situations however, we have to face nicer distributions. A natural question that arises is: do the usual RIC algorithms automatically adapt when the point samples are nicely distributed. We answer positively to this question for the case of the Delaunay triangulation of $ϵ$-nets.

$ϵ$-nets are a class of nice distributions in which the point set is such that any ball of radius $ϵ$ contains at least one point of the net and two points of the net are distance at least $ϵ$ apart. The Delaunay triangulations of $ϵ$-nets are proved to have linear size; unfortunately this is not enough to ensure a good time complexity of the randomized incremental construction of the Delaunay triangulation. In [33], [38], we prove that a uniform random sample of a given size that is taken from an $ϵ$-net has a linear sized Delaunay triangulation in any dimension. This result allows us to prove that the randomized incremental construction needs an expected linear size and an expected $O(nlogn)$ time.

Further, we also prove similar results in the case of non-Euclidean metrics, when the point distribution satisfies a certain bounded expansion property; such metrics can occur, for example, when the points are distributed on a low-dimensional manifold in a high-dimensional ambient space.

Kernelization of the Subset General Position problem in Geometry

Participants : Jean-Daniel Boissonnat, Kunal Dutta.

In collaboration with Arijit Ghosh (Indian Statistical Institute) and Sudeshna Kolay (Eindhoven University of Technology).

In [21], we consider variants of the Geometric Subset General Position problem. In defining this problem, a geometric subsystem is specified, like a subsystem of lines, hyperplanes or spheres. The input of the problem is a set of $n$ points in ${ℝ}^d$ and a positive integer $k$. The objective is to find a subset of at least $k$ input points such that this subset is in general position with respect to the specified subsystem. For example, a set of points is in general position with respect to a subsystem of hyperplanes in ${ℝ}^d$ if no $d+1$ points lie on the same hyperplane. In this paper, we study the Hyperplane Subset General Position problem under two parameterizations. When parameterized by $k$ then we exhibit a polynomial kernelization for the problem. When parameterized by $h=n-k$, or the dual parameter, then we exhibit polynomial kernels which are also tight, under standard complexity theoretic assumptions. We can also conclude similar kernelization results for d-Polynomial Subset General Position , where a vector space of polynomials of degree at most $d$ are specified as the underlying subsystem such that the size of the basis for this vector space is $b$. The objective is to find a set of at least $k$ input points, or in the dual delete at most $h=n-k$ points, such that no $b+1$ points lie on the same polynomial. Notice that this is a generalization of many well-studied geometric variants of the Set Cover problem, such as Circle Subset General Position . We also study the general projective variants of these problems. These problems are also related to other geometric problems like Subset Delaunay Triangulation problem.

Tight Kernels for Covering and Hitting: Point Hyperplane Cover and Polynomial Point Hitting Set

Participants : Jean-Daniel Boissonnat, Kunal Dutta.

In collaboration with Arijit Ghosh (Indian Statistical Institute) and Sudeshna Kolay (Eindhoven University of Technology).

The Point Hyperplane Cover problem in ${ℝ}^d$ takes as input a set of $n$ points in ${ℝ}^d$ and a positive integer $k$. The objective is to cover all the given points with a set of at most $k$ hyperplanes. The $D$-Polynomial Points Hitting Set ($D$-Polynomial Points HS ) problem in ${ℝ}^d$ takes as input a family $ℱ$ of $D$-degree polynomials from a vector space $ℛ$ in ${ℝ}^d$, and determines whether there is a set of at most $k$ points in ${ℝ}^d$ that hit all the polynomials in $ℱ$. In [22], we exhibit tight kernels where $k$ is the parameter for these problems.

Shallow packings, semialgebraic set systems, Macbeath regions, and polynomial partitioning

Participant : Kunal Dutta.

In collaboration with Arijit Ghosh (Indian Statistical Institute) and Bruno Jartoux (Université Paris-Est, Laboratoire d'Informatique Gaspard-Monge, ESIEE Paris, France) and Nabil H. Mustafa (Université Paris-Est, Laboratoire d'Informatique Gaspard-Monge, ESIEE Paris, France).

The packing lemma of Haussler states that given a set system $(X,ℝ)$ with bounded VC dimension, if every pair of sets in $ℝ$ have large symmetric difference, then $ℝ$ cannot contain too many sets. Recently it was generalized to the shallow packing lemma, applying to set systems as a function of their shallow-cell complexity. In [29] we present several new results and applications related to packings:

1. an optimal lower bound for shallow packings,

2. improved bounds on Mnets, providing a combinatorial analogue to Macbeath regions in convex geometry,

3. we observe that Mnets provide a general, more powerful framework from which the state-of-the-art unweighted $ϵ$-net results follow immediately, and

4. simplifying and generalizing one of the main technical tools in Fox et al. (J. of the EMS, to appear).

A Simple Proof of Optimal Epsilon Nets

Participant : Kunal Dutta.

In collaboration with Nabil H. Mustafa (Université Paris-Est, Laboratoire d'Informatique Gaspard-Monge, ESIEE Paris, France, and Arijit Ghosh (Indian Statistical Institute) ).

Showing the existence of $ϵ$-nets of small size has been the subject of investigation for almost 30 years, starting from the initial breakthrough of Haussler and Welzl (1987). Following a long line of successive improvements, recent results have settled the question of the size of the smallest $ϵ$-nets for set systems as a function of their so-called shallow-cell complexity.

In [20] we give a short proof of this theorem in the space of a few elementary paragraphs, showing that it follows by combining the $ϵ$-net bound of Haussler and Welzl (1987) with a variant of Haussler's packing lemma (1991).

This implies all known cases of results on unweighted $ϵ$-nets studied for the past 30 years, starting from the result of Matoušek, Seidel and Welzl (1990) to that of Clarkson and Varadajan (2007) to that of Varadarajan (2010) and Chan, Grant, Könemann and Sharpe (2012) for the unweighted case, as well as the technical and intricate paper of Aronov, Ezra and Sharir (2010).

On Subgraphs of Bounded Degeneracy in Hypergraphs

Participant : Kunal Dutta.

In collaboration with Arijit Ghosh (Indian Statistical Institute) ).

A $k$-uniform hypergraph is $d$-degenerate if every induced subgraph has a vertex of degree at most $d$. In [48], given a $k$-uniform hypergraph $H=\left(V\right(H),E(H\left)\right)$, we show there exists an induced subgraph of size at least

where $c_k=2^{-\left(1+\frac{1}{k-1}\right)}\left(1-\frac{1}{k}\right)$ and $d_H\left(v\right)$ denotes the degree of vertex $v$ in the hypergraph $H$. This connects, extends, and generalizes results of Alon-Kahn-Seymour (1987), on $d$-degenerate sets of graphs, Dutta-Mubayi-Subramanian (2012) on $d$-degenerate sets of linear hypergraphs, and Srinivasan-Shachnai (2004) on independent sets in hypergraphs to $d$-degenerate subgraphs of hypergraphs. Our technique also gives optimal lower bounds for a more generalized definition of degeneracy introduced by Zaker (2013). We further give a simple non-probabilistic proof of the Dutta-Mubayi-Subramanian bound for linear $k$-uniform hypergraphs, which extends the Alon-Kahn-Seymour proof technique to hypergraphs. Finally we provide several applications in discrete geometry, extending results of Payne-Wood (2013) and Cardinal-Tóth-Wood (2016). We also address some natural algorithmic questions. The proof of our main theorem combines the random permutation technique of Bopanna-Caro-Wei and Beame and Luby, together with a new local density argument which may be of independent interest.