1,878 research outputs found
Deformed Statistics Kullback-Leibler Divergence Minimization within a Scaled Bregman Framework
The generalized Kullback-Leibler divergence (K-Ld) in Tsallis statistics
[constrained by the additive duality of generalized statistics (dual
generalized K-Ld)] is here reconciled with the theory of Bregman divergences
for expectations defined by normal averages, within a measure-theoretic
framework. Specifically, it is demonstrated that the dual generalized K-Ld is a
scaled Bregman divergence. The Pythagorean theorem is derived from the minimum
discrimination information-principle using the dual generalized K-Ld as the
measure of uncertainty, with constraints defined by normal averages. The
minimization of the dual generalized K-Ld, with normal averages constraints, is
shown to exhibit distinctly unique features.Comment: 16 pages. Iterative corrections and expansion
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
Spectral Sparsification and Regret Minimization Beyond Matrix Multiplicative Updates
In this paper, we provide a novel construction of the linear-sized spectral
sparsifiers of Batson, Spielman and Srivastava [BSS14]. While previous
constructions required running time [BSS14, Zou12], our
sparsification routine can be implemented in almost-quadratic running time
.
The fundamental conceptual novelty of our work is the leveraging of a strong
connection between sparsification and a regret minimization problem over
density matrices. This connection was known to provide an interpretation of the
randomized sparsifiers of Spielman and Srivastava [SS11] via the application of
matrix multiplicative weight updates (MWU) [CHS11, Vis14]. In this paper, we
explain how matrix MWU naturally arises as an instance of the
Follow-the-Regularized-Leader framework and generalize this approach to yield a
larger class of updates. This new class allows us to accelerate the
construction of linear-sized spectral sparsifiers, and give novel insights on
the motivation behind Batson, Spielman and Srivastava [BSS14]
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