Abstract
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 -Jensen–Bregman divergences and derive thereof the vector-skew -Jensen–Shannon divergences. We prove that the vector-skew -Jensen–Shannon divergences are f-divergences and study the properties of these novel 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.
Highlights
Let (X, F, μ) be a measure space [1] where X denotes the sample space, F the σ-algebra of measurable events, and μ a positive measure; for example, the measure space defined by the Lebesgue measure μ L with Borel σ-algebra B(Rd ) for X = Rd or the measure space defined by the counting measure μc with the power set σ-algebra 2X on a finite alphabet X
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
We prove that weighted vector-skew Jensen–Shannon divergences are f -divergences (Theorem 1), and show how to build families of symmetric Jensen–Shannon-type divergences which can be controlled by a vector of parameters in Section 2.3, generalizing the work of [20] from scalar skewing to vector skewing
Summary
The asymmetric α-skew Jensen–Shannon divergence can be defined for a scalar parameter α ∈ (0, 1) by considering the weighted mixture ( pq)α as follows: JSαa ( p : q). This yields a generalization of the symmetric skew α-Jensen–Shannon divergences to a vector-skew parameter This extension retains the key properties for being upper-bounded and for application to densities with potentially different supports. We prove that weighted vector-skew Jensen–Shannon divergences are f -divergences (Theorem 1), and show how to build families of symmetric Jensen–Shannon-type divergences which can be controlled by a vector of parameters, generalizing the work of [20] from scalar skewing to vector skewing This may prove useful in applications by providing additional tuning parameters (which can be set, for example, by using cross-validation techniques). The experimental results graphically compare the Jeffreys centroid with the Jensen–Shannon centroid for grey-valued image histograms
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