An Upper Bound on the Average Size of Silhouettes

It is a widely observed phenomenon in computer graphics that the size of the silhouette of a polyhedron is much smaller than the size of the whole polyhedron. This paper provides, for the first time, theoretical evidence supporting this for a large class of objects, namely for polyhedra that approximate surfaces in some reasonable way; the surfaces may be non-convex and non-differentiable and they may have boundaries. We prove that such polyhedra have silhouettes of expected size $O(\sqrt{n})$ where the average is taken over all points of view and n is the complexity of the polyhedron

Optimal rates of convergence for persistence diagrams in Topological Data Analysis

Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field. In this paper, we study topological persistence in general metric spaces, with a statistical approach. We show that the use of persistent homology can be naturally considered in general statistical frameworks and persistence diagrams can be used as statistics with interesting convergence properties. Some numerical experiments are performed in various contexts to illustrate our results

The structure and stability of persistence modules

We give a self-contained treatment of the theory of persistence modules indexed over the real line. We give new proofs of the standard results. Persistence diagrams are constructed using measure theory. Linear algebra lemmas are simplified using a new notation for calculations on quiver representations. We show that the stringent finiteness conditions required by traditional methods are not necessary to prove the existence and stability of the persistence diagram. We introduce weaker hypotheses for taming persistence modules, which are met in practice and are strong enough for the theory still to work. The constructions and proofs enabled by our framework are, we claim, cleaner and simpler.Comment: New version. We discuss in greater depth the interpolation lemma for persistence module

On the Smoothed Complexity of Convex Hulls

We establish an upper bound on the smoothed complexity of convex hulls in R^d under uniform Euclidean (L^2) noise. Specifically, let {p_1^*, p_2^*, ..., p_n^*} be an arbitrary set of n points in the unit ball in R^d and let p_i = p_i^* + x_i, where x_1, x_2, ..., x_n are chosen independently from the unit ball of radius r. We show that the expected complexity, measured as the number of faces of all dimensions, of the convex hull of {p_1, p_2, ..., p_n} is O(n^{2-4/(d+1)} (1+1/r)^{d-1}); the magnitude r of the noise may vary with n. For d=2 this bound improves to O(n^{2/3} (1+r^{-2/3})). We also analyze the expected complexity of the convex hull of L^2 and Gaussian perturbations of a nice sample of a sphere, giving a lower-bound for the smoothed complexity. We identify the different regimes in terms of the scale, as a function of n, and show that as the magnitude of the noise increases, that complexity varies monotonically for Gaussian noise but non-monotonically for L^2 noise

The monotonicity of f-vectors of random polytopes

Let K be a compact convex body in Rd, let Kn be the convex hull of n points chosen uniformly and independently in K, and let fi(Kn) denote the number of i-dimensional faces of Kn. We show that for planar convex sets, E(f0(Kn)) is increasing in n. In dimension d>=3 we prove that if lim(E((f[d -1](Kn))/(An^c)->1 when n->infinity for some constants A and c > 0 then the function E(f[d-1](Kn)) is increasing for n large enough. In particular, the number of facets of the convex hull of n random points distributed uniformly and independently in a smooth compact convex body is asymptotically increasing. Our proof relies on a random sampling argument

Complexity analysis of random geometric structures made simpler

Average-case analysis of data-structures or algorithms is commonly used in compu- tational geometry when the, more classical, worst-case analysis is deemed overly pessimistic. Since these analyses are often intricate, the models of random geometric data that can be handled are often simplistic and far from "realistic inputs". We present a new simple scheme for the analy- sis of geometric structures. While this scheme only produces results up to a polylog factor, it is much simpler to apply than the classical techniques and therefore succeeds in analyzing new input distributions related to smoothed complexity analysis. Abstract: We illustrate our method on two classical structures: convex hulls and Delaunay triangulations. Specifically, we give short and elementary proofs of the classical results that n points uniformly distributed in a ball in Rd have a convex hull and a Delaunay triangulation of respective expected complexities Θ~(n^((d+1)/(d-1)) ) and Θ~(n). We then prove that if we start with n points well-spread on a sphere, e.g. an (ε,κ)-sample of that sphere, and perturb that sample by moving each point ran- domly and uniformly within distance at most δ of its initial position, then the expected complexity of the convex hull of the resulting point set is Θ~( sqrt(n)^(1−1/d) δ^(-(d-1/d)/4)). .L'analyse en moyenne de structure de données et d'algorithmes géométriques est fréquemment utilisée en géométrie algorithmique, un domaine ou' l'analyse dans le cas le pire est souvent très pessimiste. La difficulté de ce type d'analyse fait que les modèles probabilistes utilisés restent simples et relativement éloignées de données réalistes. Nous présentons une nouvelle approche pour l'analyse des structures géométriques. Nos résultats sont seulement 'a des facteurs logarithmiques près, mais notre méthode est plus simple que les classiques du domaine et nous réussissons 'a analyser de nouveau type de distribution liée à la smooth analysis. Nous illustrons notre méthode sur deux structures classiques: l'enveloppe convexe et la triangulation de Delaunay. Plus précisément, nous démontrons simplement le fait, classique, que n points uniformément distribués dans une boule de Rd ont une enveloppe convexe et une triangulation de Delaunay dont l'espérance de la taille est respectivement Θ~(n^((d+1)/(d-1)) ) et Θ~(n). Nous démontrons ensuite que si on se donne ensemble de n points bien distribu ́es sur une sphère, par exemple un (ε, κ)-échantillon de la sphère, et qu'on le perturbe ensuite en déplaçant chaque point uniformément d'une distance δ à partir de sa position initiale, alors l'espérance de la taille de l'enveloppe convexe de ces points est Θ~( sqrt(n)^(1−1/d) δ^(-(d-1/d)/4)).

DTM-based Filtrations

Despite strong stability properties, the persistent homology of filtrations classically used in Topological Data Analysis, such as, e.g. the Cech or Vietoris-Rips filtrations, are very sensitive to the presence of outliers in the data from which they are computed. In this paper, 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.Comment: Abel Symposia, Springer, In press, Topological Data Analysi

Recognizing shrinkable complexes is NP-complete

International audienceWe say that a simplicial complex is shrinkable if there exists a sequence of admissible edge contractions that reduces the complex to a single vertex. We prove that it is NP-complete to decide whether a (three-dimensional) simplicial complex is shrinkable. Along the way, we describe examples of contractible complexes which are not shrinkable

On Order Types of Random Point Sets

A simple method to produce a random order type is to take the order type of a random point set. We conjecture that many probability distributions on order types defined in this way are heavily concentrated and therefore sample inefficiently the space of order types. We present two results on this question. First, we study experimentally the bias in the order types of $n$ random points chosen uniformly and independently in a square, for $n$ up to $16$. Second, we study algorithms for determining the order type of a point set in terms of the number of coordinate bits they require to know. We give an algorithm that requires on average $4n \log\_2 n+O(n)$ bits to determine the order type of $P$, and show that any algorithm requires at least $4n \log\_2 n - O(n \log\log n)$ bits. This implies that the concentration conjecture cannot be proven by an "efficient encoding" argument

La triangulation de Delaunay d'un échantillon aléatoire d'un bon échantillon a une taille linéaire

A good sample is a point set such that any ball of radius $\epsilon$ contains a constant number of points. The Delaunay triangulation of a good sample is 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 this paper we prove that a random Bernoulli sample of a good sample has a triangulation of linear size. This result allows to prove that the randomized incremental construction needs an expected linear size and an expected $O(n\log n)$ time.Un bon échantillon est un ensemble de points tel que toute boule de rayon $\epsilon$ contienne un nombre constant de points.Il est démontré que la triangulation de Delaunay d'un bon échantillon a une taille linéaire, malheureusement cela ne suffit pas à assurerune bonne complexité à la construction incrémentale randomisée de latriangulation de Delaunay.Dans ce rapport, nous démontrons que la triangulation d'un échantillon aléatoire de Bernoullid'un bon échantillon a une taille linéaire. Nous en déduisonsque la construction incrémentale randomisée peut être faite en temps$O(n \log n)$ et espace $O(n)$