Total Jensen divergences: Definition, Properties and k-Means++ Clustering

We present a novel class of divergences induced by a smooth convex function called total Jensen divergences. Those total Jensen divergences are invariant by construction to rotations, a feature yielding regularization of ordinary Jensen divergences by a conformal factor. We analyze the relationships between this novel class of total Jensen divergences and the recently introduced total Bregman divergences. We then proceed by defining the total Jensen centroids as average distortion minimizers, and study their robustness performance to outliers. Finally, we prove that the k-means++ initialization that bypasses explicit centroid computations is good enough in practice to guarantee probabilistically a constant approximation factor to the optimal k-means clustering.Comment: 27 page

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

An equivalent condition to the Jensen inequality for the generalized Sugeno integral.

For the classical Jensen inequality of convex functions, i.e., [Formula: see text] an equivalent condition is proved in the framework of the generalized Sugeno integral. Also, the necessary and sufficient conditions for the validity of the discrete form of the Jensen inequality for the generalized Sugeno integral are given

Transcript of The Blimp Encounter

This story is an excerpt from a longer interview that was collected as part of the Launching through the Surf: The Dory Fleet of Pacific City project. In this story, Al Jensen recounts the experience of encountering a Navy blimp while fishing with his father, Pete Jensen, during World War II

Properties of Classical and Quantum Jensen-Shannon Divergence

Jensen-Shannon divergence (JD) is a symmetrized and smoothed version of the most important divergence measure of information theory, Kullback divergence. As opposed to Kullback divergence it determines in a very direct way a metric; indeed, it is the square of a metric. We consider a family of divergence measures (JD_alpha for alpha>0), the Jensen divergences of order alpha, which generalize JD as JD_1=JD. Using a result of Schoenberg, we prove that JD_alpha is the square of a metric for alpha lies in the interval (0,2], and that the resulting metric space of probability distributions can be isometrically embedded in a real Hilbert space. Quantum Jensen-Shannon divergence (QJD) is a symmetrized and smoothed version of quantum relative entropy and can be extended to a family of quantum Jensen divergences of order alpha (QJD_alpha). We strengthen results by Lamberti et al. by proving that for qubits and pure states, QJD_alpha^1/2 is a metric space which can be isometrically embedded in a real Hilbert space when alpha lies in the interval (0,2]. In analogy with Burbea and Rao's generalization of JD, we also define general QJD by associating a Jensen-type quantity to any weighted family of states. Appropriate interpretations of quantities introduced are discussed and bounds are derived in terms of the total variation and trace distance.Comment: 13 pages, LaTeX, expanded contents, added references and corrected typo