kk-MLE: A fast algorithm for learning statistical mixture models

We describe $k$-MLE, a fast and efficient local search algorithm for learning finite statistical mixtures of exponential families such as Gaussian mixture models. Mixture models are traditionally learned using the expectation-maximization (EM) soft clustering technique that monotonically increases the incomplete (expected complete) likelihood. Given prescribed mixture weights, the hard clustering $k$-MLE algorithm iteratively assigns data to the most likely weighted component and update the component models using Maximum Likelihood Estimators (MLEs). Using the duality between exponential families and Bregman divergences, we prove that the local convergence of the complete likelihood of $k$-MLE follows directly from the convergence of a dual additively weighted Bregman hard clustering. The inner loop of $k$-MLE can be implemented using any $k$-means heuristic like the celebrated Lloyd's batched or Hartigan's greedy swap updates. We then show how to update the mixture weights by minimizing a cross-entropy criterion that implies to update weights by taking the relative proportion of cluster points, and reiterate the mixture parameter update and mixture weight update processes until convergence. Hard EM is interpreted as a special case of $k$-MLE when both the component update and the weight update are performed successively in the inner loop. To initialize $k$-MLE, we propose $k$-MLE++, a careful initialization of $k$-MLE guaranteeing probabilistically a global bound on the best possible complete likelihood.Comment: 31 pages, Extend preliminary paper presented at IEEE ICASSP 201

Non-Euclidean Erd\H{o}s-Anning Theorems

The Erd\H{o}s-Anning theorem states that every point set in the Euclidean plane with integer distances must be either collinear or finite. More strongly, for any (non-degenerate) triangle of diameter $\delta$, at most $O(\delta^2)$ points can have integer distances from all three triangle vertices. We prove the same results for any strictly convex distance function on the plane, and analogous results for every two-dimensional complete Riemannian manifold of bounded genus and for geodesic distance on the boundary of every three-dimensional Euclidean convex set. As a consequence, we resolve a 1983 question of Richard Guy on the equilateral dimension of Riemannian manifolds. Our proofs are based on the properties of additively weighted Voronoi diagrams of these distances.Comment: 26 pages, 13 figures. This version adds a connection to equilateral dimensio

Districting Problems - New Geometrically Motivated Approaches

This thesis focuses on districting problems were the basic areas are represented by points or lines. In the context of points, it presents approaches that utilize the problem\u27s underlying geometrical information. For lines it introduces an algorithm combining features of geometric approaches, tabu search, and adaptive randomized neighborhood search that includes the routing distances explicitly. Moreover, this thesis summarizes, compares and enhances existing compactness measures

Piecewise-Linear Farthest-Site Voronoi Diagrams

Voronoi diagrams induced by distance functions whose unit balls are convex polyhedra are piecewise-linear structures. Nevertheless, analyzing their combinatorial and algorithmic properties in dimensions three and higher is an intriguing problem. The situation turns easier when the farthest-site variants of such Voronoi diagrams are considered, where each site gets assigned the region of all points in space farthest from (rather than closest to) it. We give asymptotically tight upper and lower worst-case bounds on the combinatorial size of farthest-site Voronoi diagrams for convex polyhedral distance functions in general dimensions, and propose an optimal construction algorithm. Our approach is uniform in the sense that (1) it can be extended from point sites to sites that are convex polyhedra, (2) it covers the case where the distance function is additively and/or multiplicatively weighted, and (3) it allows an anisotropic scenario where each site gets allotted its particular convex distance polytope

Vorosweep: a fast generalized crystal growing Voronoi diagram generation algorithm

We propose a new algorithm for generating quickly approximate generalized Voronoi diagrams of point sites associated to arbitrary convex distance metric in the Euclidian plane. This algorithm produces connected cells by emulating the growth of crystals starting at the point sites, in order to reduce the complexity of the diagram. The main practical contribution is the Vorosweep package which is the reference implementation of the algorithm. Experimental results and benchmarks are given to demonstrate the versatility of this approach.WIST 3 grant 1017074 DOMHEX (Dominant Hexahedral Mesh Generation

Higher-order Voronoi diagrams of polygonal objects

Higher-order Voronoi diagrams are fundamental geometric structures which encode the k-nearest neighbor information. Thus, they aid in computations that require proximity information beyond the nearest neighbor. They are related to various favorite structures in computational geometry and are a fascinating combinatorial problem to study. While higher-order Voronoi diagrams of points have been studied a lot, they have not been considered for other types of sites. Points lack dimensionality which makes them unable to represent various real-life instances. Points are the simplest kind of geometric object and therefore higher- order Voronoi diagrams of points can be considered as the corner case of all higher-order Voronoi diagrams. The goal of this dissertation is to move away from the corner and bring the higher-order Voronoi diagram to more general geometric instances. We focus on certain polygonal objects as they provide flexibility and are able to represent real-life instances. Before this dissertation, higher-order Voronoi diagrams of polygonal objects had been studied only for the nearest neighbor and farthest Voronoi diagrams. In this dissertation we investigate structural and combinatorial properties and discover that the dimensionality of geometric objects manifests itself in numerous ways which do not exist in the case of points. We prove that the structural complexity of the order-k Voronoi diagram of non-crossing line segments is O(k(n-k)), as in the case of points. We study disjoint line segments, intersecting line segments, line segments forming a planar straight-line graph and extend the results to the Lp metric, 1<=p<=infty. We also establish the connection between two mathematical abstractions: abstract Voronoi diagrams and the Clarkson-Shor framework. We design several construction algorithms that cover the case of non-point sites. While computational geometry provides several approaches to study the structural complexity that give tight realizable bounds, developing an effective construction algorithm is still a challenging problem even for points. Most of the construction algorithms are designed to work with points as they utilize their simplicity and relations with data-structures that work specifically for points. We extend the iterative and the sweepline approaches that are quite efficient in constructing all order-i Voronoi diagrams, for i<=k and we also give three randomized construction algorithms for abstract higher-order Voronoi diagrams that deal specifically with the construction of the order-k Voronoi diagrams

Curved Voronoi diagrams

Voronoi diagrams are fundamental data structures that have been extensively studied in Computational Geometry. A Voronoi diagram can be defined as the minimization diagram of a finite set of continuous functions. Usually, each of those functions is interpreted as the distance function to an object. The as- sociated Voronoi diagram subdivides the embedding space into regions, each region consisting of the points that are closer to a given object than to the others. We may define many variants of Voronoi diagrams depending on the class of objects, the distance functions and the embedding space. Affine di- agrams, i.e. diagrams whose cells are convex polytopes, are well understood. Their properties can be deduced from the properties of polytopes and they can be constructed efficiently. The situation is very different for Voronoi dia- grams with curved regions. Curved Voronoi diagrams arise in various contexts where the objects are not punctual or the distance is not the Euclidean dis- tance. We survey the main results on curved Voronoi diagrams. We describe in some detail two general mechanisms to obtain effective algorithms for some classes of curved Voronoi diagrams. The first one consists in linearizing the diagram and applies, in particular, to diagrams whose bisectors are algebraic hypersurfaces. The second one is a randomized incremental paradigm that can construct affine and several planar non-affine diagrams. We finally introduce the concept of Medial Axis which generalizes the concept of Voronoi diagram to infinite sets. Interestingly, it is possible to efficiently construct a certified approximation of the medial axis of a bounded set from the Voronoi diagram of a sample of points on the boundary of the set