134 research outputs found
-MLE: A fast algorithm for learning statistical mixture models
We describe -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 -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 -MLE follows directly from the convergence of a dual
additively weighted Bregman hard clustering. The inner loop of -MLE can be
implemented using any -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 -MLE when both the component update and
the weight update are performed successively in the inner loop. To initialize
-MLE, we propose -MLE++, a careful initialization of -MLE guaranteeing
probabilistically a global bound on the best possible complete likelihood.Comment: 31 pages, Extend preliminary paper presented at IEEE ICASSP 201
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
Bregman Voronoi diagrams
A preliminary version appeared in the 18th ACM-SIAM Symposium on Discrete Algorithms, pp. 746- 755, 2007International audienceThe 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 various variants of Voronoi diagrams depending on the class of objects, the distance function 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 one to define information-theoretic Voronoi diagrams in sta- tistical 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 connection with Bregman Voronoi diagrams. We show that these triangulations capture many of the properties of the celebrated Delaunay triangulation
The Hidden Geometry of Particle Collisions
We establish that many fundamental concepts and techniques in quantum field
theory and collider physics can be naturally understood and unified through a
simple new geometric language. The idea is to equip the space of collider
events with a metric, from which other geometric objects can be rigorously
defined. Our analysis is based on the energy mover's distance, which quantifies
the "work" required to rearrange one event into another. This metric, which
operates purely at the level of observable energy flow information, allows for
a clarified definition of infrared and collinear safety and related concepts. A
number of well-known collider observables can be exactly cast as the minimum
distance between an event and various manifolds in this space. Jet definitions,
such as exclusive cone and sequential recombination algorithms, can be directly
derived by finding the closest few-particle approximation to the event. Several
area- and constituent-based pileup mitigation strategies are naturally
expressed in this formalism as well. Finally, we lift our reasoning to develop
a precise distance between theories, which are treated as collections of events
weighted by cross sections. In all of these various cases, a better
understanding of existing methods in our geometric language suggests
interesting new ideas and generalizations.Comment: 56 pages, 11 figures, 5 tables; v2: minor changes and updated
references; v3: updated to match JHEP versio
Algorithmic Superactivation of Asymptotic Quantum Capacity of Zero-Capacity Quantum Channels
The superactivation of zero-capacity quantum channels makes it possible to
use two zero-capacity quantum channels with a positive joint capacity for their
output. Currently, we have no theoretical background to describe all possible
combinations of superactive zero-capacity channels; hence, there may be many
other possible combinations. In practice, to discover such superactive
zero-capacity channel-pairs, we must analyze an extremely large set of possible
quantum states, channel models, and channel probabilities. There is still no
extremely efficient algorithmic tool for this purpose. This paper shows an
efficient algorithmical method of finding such combinations. Our method can be
a very valuable tool for improving the results of fault-tolerant quantum
computation and possible communication techniques over very noisy quantum
channels.Comment: 35 pages, 17 figures, Journal-ref: Information Sciences (Elsevier,
2012), presented in part at Quantum Information Processing 2012 (QIP2012),
v2: minor changes, v3: published version; Information Sciences, Elsevier,
ISSN: 0020-0255; 201
Une fonction distance à k points pour l'inférence géométrique robuste
Analyzing the sub-level sets of the distance to a compact sub-manifold of R d is a common method in topological data analysis, to understand its topology. Therefore, topological inference procedures usually rely on a distance estimate based on n sample points [41]. In the case where sample points are corrupted by noise, the distance-to-measure function (DTM, [16]) is a surrogate for the distance-to-compact-set function. In practice, computing the homology of its sub-level sets requires to compute the homology of unions of n balls ([28, 14]), that might become intractable whenever n is large. To simultaneously face the two problems of a large number of points and noise, we introduce the k-power-distance-to-measure function (k-PDTM). This new approximation of the distance-to-measure may be thought of as a k-pointbased approximation of the DTM. Its sublevel sets consist in unions of k balls, and this distance is also proved robust to noise. We assess the quality of this approximation for k possibly drastically smaller than n, and provide an algorithm to compute this k-PDTM from a sample. Numerical experiments illustrate the good behavior of this k-points approximation in a noisy topological inference framework.Afin de comprendre la topologie d'une sous-variété compacte de R^d, il est courant en analyse topologique des données d'analyser les sous-niveaux de la fonction distance à cette sous-variété. C'est pourquoi, les procédures d'inférence topologique reposent souvent sur des estimées de la fonction distance, construites sur n points. Lorsque l'échantillon de points est corrompu par des données aberrantes, la fonction distance à la mesure (DTM) est une alternative à la distance au compact. En pratique, le calcul de l'homologie de ses sous-niveaux revient à calculer l'homologie d'unions de n boules, ce qui devient impossible lorsque n est grand. Afin de pallier simultanément le problème du grand nombre de points et du bruit, nous introduisons la fonction k-puissance distance à la mesure (k-PDTM). Il s'agit d'une nouvelle approximation de la fonction distance à la mesure qui peut être vue comme une approximation de la DTM basée sur k points. Ses sous-niveaux sont des unions de k boules, et cette distance est robuste au bruit. Nous étudions la qualité de cette approximation pour k possiblement beaucoup plus petit que n, et fournissons un algorithme permettant de calculer cette k-PDTM à partir d'un échantillon de points. Des expériences numériques illustrent le bon comportement de cette approximation construite sur k points, dans le cadre de l'inférence topologique avec bruit
A unified approach to energy conservation and potential vorticity dynamics for arbitrarily-structured C-grids
Copyright © 2010 Elsevier. NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Computational Physics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Computational Physics, Vol. 229, Issue 9 (2010), DOI: 10.1016/j.jcp.2009.12.007A numerical scheme applicable to arbitrarily-structured C-grids is presented for the nonlinear shallow-water equations. By discretizing the vector-invariant form of the momentum equation, the relationship between the nonlinear Coriolis force and the potential vorticity flux can be used to guarantee that mass, velocity and potential vorticity evolve in a consistent and compatible manner. Underpinning the consistency and compatibility of the discrete system is the construction of an auxiliary thickness equation that is staggered from the primary thickness equation and collocated with the vorticity field. The numerical scheme also exhibits conservation of total energy to within time-truncation error. Simulations of the standard shallow-water test cases confirm the analysis and show convergence rates between 1st1st- and 2nd2nd-order accuracy when discretizing the system with quasi-uniform spherical Voronoi diagrams. The numerical method is applicable to a wide class of meshes, including latitude–longitude grids, Voronoi diagrams, Delaunay triangulations and conformally-mapped cubed-sphere meshes
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