On a generalization of the Jensen-Shannon divergence and the JS-symmetrization of distances relying on abstract means

The Jensen-Shannon divergence is a renown bounded symmetrization of the unbounded Kullback-Leibler divergence which measures the total Kullback-Leibler divergence to the average mixture distribution. However the Jensen-Shannon divergence between Gaussian distributions is not available in closed-form. To bypass this problem, we present a generalization of the Jensen-Shannon (JS) divergence using abstract means which yields closed-form expressions when the mean is chosen according to the parametric family of distributions. More generally, we define the JS-symmetrizations of any distance using generalized statistical mixtures derived from abstract means. In particular, we first show that the geometric mean is well-suited for exponential families, and report two closed-form formula for (i) the geometric Jensen-Shannon divergence between probability densities of the same exponential family, and (ii) the geometric JS-symmetrization of the reverse Kullback-Leibler divergence. As a second illustrating example, we show that the harmonic mean is well-suited for the scale Cauchy distributions, and report a closed-form formula for the harmonic Jensen-Shannon divergence between scale Cauchy distributions. We also define generalized Jensen-Shannon divergences between matrices (e.g., quantum Jensen-Shannon divergences) and consider clustering with respect to these novel Jensen-Shannon divergences.Comment: 30 page

The {\alpha}-divergences associated with a pair of strictly comparable quasi-arithmetic means

We generalize the family of $\alpha$-divergences using a pair of strictly comparable weighted means. In particular, we obtain the $1$-divergence in the limit case $\alpha\rightarrow 1$ (a generalization of the Kullback-Leibler divergence) and the $0$-divergence in the limit case $\alpha\rightarrow 0$ (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 $\alpha$-divergences and its subfamily of bipower homogeneous $\alpha$-divergences which belong to the Csis\'ar's $f$-divergences. Finally, we show that these generalized quasi-arithmetic $1$-divergences and $0$-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

On a generalization of the Jensen-Shannon divergence

The Jensen-Shannon divergence is a renown bounded symmetrization of the Kullback-Leibler divergence which does not require probability densities to have matching supports. In this paper, we introduce a vector-skew generalization of the scalar $\alpha$-Jensen-Bregman divergences and derive thereof the vector-skew $\alpha$-Jensen-Shannon divergences. We study the properties of these novel divergences and show how to build parametric families of symmetric Jensen-Shannon-type divergences. Finally, we report an iterative algorithm to numerically compute the Jensen-Shannon-type centroids for a set of probability densities belonging to a mixture family: This includes the case of the Jensen-Shannon centroid of a set of categorical distributions or normalized histograms.Comment: 19 pages, 3 figure

On Clustering Histograms with k-Means by Using Mixed α-Divergences

Clustering sets of histograms has become popular thanks to the success of the generic method of bag-of-X used in text categorization and in visual categorization applications. In this paper, we investigate the use of a parametric family of distortion measures, called the α-divergences, for clustering histograms. Since it usually makes sense to deal with symmetric divergences in information retrieval systems, we symmetrize the α -divergences using the concept of mixed divergences. First, we present a novel extension of k-means clustering to mixed divergences. Second, we extend the k-means++ seeding to mixed α-divergences and report a guaranteed probabilistic bound. Finally, we describe a soft clustering technique for mixed α-divergences

Information geometry

This Special Issue of the journal Entropy, titled “Information Geometry I”, contains a collection of 17 papers concerning the foundations and applications of information geometry. Based on a geometrical interpretation of probability, information geometry has become a rich mathematical field employing the methods of differential geometry. It has numerous applications to data science, physics, and neuroscience. Presenting original research, yet written in an accessible, tutorial style, this collection of papers will be useful for scientists who are new to the field, while providing an excellent reference for the more experienced researcher. Several papers are written by authorities in the field, and topics cover the foundations of information geometry, as well as applications to statistics, Bayesian inference, machine learning, complex systems, physics, and neuroscience