15 research outputs found
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 Burbea-Rao and Bhattacharyya centroids
We study the centroid with respect to the class of information-theoretic
Burbea-Rao divergences that generalize the celebrated Jensen-Shannon divergence
by measuring the non-negative Jensen difference induced by a strictly convex
and differentiable function. Although those Burbea-Rao divergences are
symmetric by construction, they are not metric since they fail to satisfy the
triangle inequality. We first explain how a particular symmetrization of
Bregman divergences called Jensen-Bregman distances yields exactly those
Burbea-Rao divergences. We then proceed by defining skew Burbea-Rao
divergences, and show that skew Burbea-Rao divergences amount in limit cases to
compute Bregman divergences. We then prove that Burbea-Rao centroids are
unique, and can be arbitrarily finely approximated by a generic iterative
concave-convex optimization algorithm with guaranteed convergence property. In
the second part of the paper, we consider the Bhattacharyya distance that is
commonly used to measure overlapping degree of probability distributions. We
show that Bhattacharyya distances on members of the same statistical
exponential family amount to calculate a Burbea-Rao divergence in disguise.
Thus we get an efficient algorithm for computing the Bhattacharyya centroid of
a set of parametric distributions belonging to the same exponential families,
improving over former specialized methods found in the literature that were
limited to univariate or "diagonal" multivariate Gaussians. To illustrate the
performance of our Bhattacharyya/Burbea-Rao centroid algorithm, we present
experimental performance results for -means and hierarchical clustering
methods of Gaussian mixture models.Comment: 13 page
Non-flat clustering whith alpha-divergences
International audienceThe scope of the well-known -means algorithm has been broadly extended with some recent results: first, the k-means++ initialization method gives some approximation guarantees; second, the Bregman k-means algorithm generalizes the classical algorithm to the large family of Bregman divergences. The Bregman seeding framework combines approximation guarantees with Bregman divergences. We present here an extension of the k-means algorithm using the family of alpha-divergences. With the framework for representational Bregman divergences, we show that an alpha-divergence based k-means algorithm can be designed. We present preliminary experiments for clustering and image segmentation applications. Since alpha-divergences are the natural divergences for constant curvature spaces, these experiments are expected to give information on the structure of the data
The {\alpha}-divergences associated with a pair of strictly comparable quasi-arithmetic means
We generalize the family of -divergences using a pair of strictly
comparable weighted means. In particular, we obtain the -divergence in the
limit case (a generalization of the Kullback-Leibler
divergence) and the -divergence in the limit case (a
generalization of the reverse Kullback-Leibler divergence). We state the
condition for a pair of quasi-arithmetic means to be strictly comparable, and
report the formula for the quasi-arithmetic -divergences and its
subfamily of bipower homogeneous -divergences which belong to the
Csis\'ar's -divergences. Finally, we show that these generalized
quasi-arithmetic -divergences and -divergences can be decomposed as the
sum of generalized cross-entropies minus entropies, and rewritten as conformal
Bregman divergences using monotone embeddings.Comment: 18 page
Precision-Recall Curves Using Information Divergence Frontiers
Despite the tremendous progress in the estimation of generative models, the
development of tools for diagnosing their failures and assessing their
performance has advanced at a much slower pace. Recent developments have
investigated metrics that quantify which parts of the true distribution is
modeled well, and, on the contrary, what the model fails to capture, akin to
precision and recall in information retrieval. In this paper, we present a
general evaluation framework for generative models that measures the trade-off
between precision and recall using R\'enyi divergences. Our framework provides
a novel perspective on existing techniques and extends them to more general
domains. As a key advantage, this formulation encompasses both continuous and
discrete models and allows for the design of efficient algorithms that do not
have to quantize the data. We further analyze the biases of the approximations
used in practice.Comment: Updated to the AISTATS 2020 versio
Centroid-Based Clustering with ab-Divergences
Centroid-based clustering is a widely used technique within unsupervised learning
algorithms in many research fields. The success of any centroid-based clustering relies on the
choice of the similarity measure under use. In recent years, most studies focused on including several
divergence measures in the traditional hard k-means algorithm. In this article, we consider the
problem of centroid-based clustering using the family of ab-divergences, which is governed by two
parameters, a and b. We propose a new iterative algorithm, ab-k-means, giving closed-form solutions
for the computation of the sided centroids. The algorithm can be fine-tuned by means of this pair of
values, yielding a wide range of the most frequently used divergences. Moreover, it is guaranteed to
converge to local minima for a wide range of values of the pair (a, b). Our theoretical contribution
has been validated by several experiments performed with synthetic and real data and exploring the
(a, b) plane. The numerical results obtained confirm the quality of the algorithm and its suitability to
be used in several practical applications.MINECO TEC2017-82807-
Centroid-Based Clustering with αβ-Divergences
Article number 196Centroid-based clustering is a widely used technique within unsupervised learning
algorithms in many research fields. The success of any centroid-based clustering relies on the
choice of the similarity measure under use. In recent years, most studies focused on including several
divergence measures in the traditional hard k-means algorithm. In this article, we consider the
problem of centroid-based clustering using the family of αβ-divergences, which is governed by two
parameters, α and β. We propose a new iterative algorithm, αβ-k-means, giving closed-form solutions
for the computation of the sided centroids. The algorithm can be fine-tuned by means of this pair of
values, yielding a wide range of the most frequently used divergences. Moreover, it is guaranteed to
converge to local minima for a wide range of values of the pair (α, β). Our theoretical contribution
has been validated by several experiments performed with synthetic and real data and exploring the
(α, β) plane. The numerical results obtained confirm the quality of the algorithm and its suitability to
be used in several practical applicationsMinisterio de Economía y Competitividad de España (MINECO) TEC2017-82807-