Bregman Voronoi Diagrams: Properties, Algorithms and Applications

The Voronoi diagram of a finite set of objects is a fundamental geometric structure that 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. In this paper, we investigate a framework for defining and building Voronoi diagrams for a broad class of distance functions called Bregman divergences. Bregman divergences include not only the traditional (squared) Euclidean distance but also various divergence measures based on entropic functions. Accordingly, Bregman Voronoi diagrams allow to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We define several types of Bregman diagrams, establish correspondences between those diagrams (using the Legendre transformation), and show how to compute them efficiently. We also introduce extensions of these diagrams, e.g. k-order and k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set of points and their connexion with Bregman Voronoi diagrams. We show that these triangulations capture many of the properties of the celebrated Delaunay triangulation. Finally, we give some applications of Bregman Voronoi diagrams which are of interest in the context of computational geometry and machine learning.Comment: Extend the proceedings abstract of SODA 2007 (46 pages, 15 figures

A Smoothed Dual Approach for Variational Wasserstein Problems

Variational problems that involve Wasserstein distances have been recently proposed to summarize and learn from probability measures. Despite being conceptually simple, such problems are computationally challenging because they involve minimizing over quantities (Wasserstein distances) that are themselves hard to compute. We show that the dual formulation of Wasserstein variational problems introduced recently by Carlier et al. (2014) can be regularized using an entropic smoothing, which leads to smooth, differentiable, convex optimization problems that are simpler to implement and numerically more stable. We illustrate the versatility of this approach by applying it to the computation of Wasserstein barycenters and gradient flows of spacial regularization functionals

Sampling Sparse Signals on the Sphere: Algorithms and Applications

We propose a sampling scheme that can perfectly reconstruct a collection of spikes on the sphere from samples of their lowpass-filtered observations. Central to our algorithm is a generalization of the annihilating filter method, a tool widely used in array signal processing and finite-rate-of-innovation (FRI) sampling. The proposed algorithm can reconstruct $K$ spikes from $(K+\sqrt{K})^2$ spatial samples. This sampling requirement improves over previously known FRI sampling schemes on the sphere by a factor of four for large $K$. We showcase the versatility of the proposed algorithm by applying it to three different problems: 1) sampling diffusion processes induced by localized sources on the sphere, 2) shot noise removal, and 3) sound source localization (SSL) by a spherical microphone array. In particular, we show how SSL can be reformulated as a spherical sparse sampling problem.Comment: 14 pages, 8 figures, submitted to IEEE Transactions on Signal Processin

New Exact and Numerical Solutions of the (Convection-)Diffusion Kernels on SE(3)

We consider hypo-elliptic diffusion and convection-diffusion on $\mathbb{R}^3 \rtimes S^2$, the quotient of the Lie group of rigid body motions SE(3) in which group elements are equivalent if they are equal up to a rotation around the reference axis. We show that we can derive expressions for the convolution kernels in terms of eigenfunctions of the PDE, by extending the approach for the SE(2) case. This goes via application of the Fourier transform of the PDE in the spatial variables, yielding a second order differential operator. We show that the eigenfunctions of this operator can be expressed as (generalized) spheroidal wave functions. The same exact formulas are derived via the Fourier transform on SE(3). We solve both the evolution itself, as well as the time-integrated process that corresponds to the resolvent operator. Furthermore, we have extended a standard numerical procedure from SE(2) to SE(3) for the computation of the solution kernels that is directly related to the exact solutions. Finally, we provide a novel analytic approximation of the kernels that we briefly compare to the exact kernels.Comment: Revised and restructure