On Hölder Projective Divergences

Abstract

Abstract:We describe a framework to build distances by measuring the tightness of inequalities and introduce the notion of proper statistical divergences and improper pseudo-divergences. We then consider the Hölder ordinary and reverse inequalities and present two novel classes of Hölder divergences and pseudo-divergences that both encapsulate the special case of the Cauchy-Schwarz divergence. We report closed-form formulas for those statistical dissimilarities when considering distributions belonging to the same exponential family provided that the natural parameter space is a cone (e.g., multivariate Gaussians) or affine (e.g., categorical distributions). Those new classes of Hölder distances are invariant to rescaling and thus do not require distributions to be normalized. Finally, we show how to compute statistical Hölder centroids with respect to those divergences and carry out center-based clustering toy experiments on a set of Gaussian distributions which demonstrate empirically that symmetrized Hölder divergences outperform the symmetric Cauchy-Schwarz divergence.

k-Means Clustering with Hölder Divergences

“…Notice that this divergence construction allows one to consider the equivalence class of scaled inequalities: λ × lhs(p, q) ≤ λ × rhs(p, q), ∀λ > 0. Following this divergence construction principle, we defined Hölder divergences based on the Hölder's inequality, and presented the basic properties of this divergence family (Nielsen et al, 2017). In this paper, we further extend the empirical clustering study with respect to Hölder divergences, and show that symmetrized Hölder divergences consistently outperform significantly the Cauchy-Schwarz divergence (Hasanbelliu et al, 2014).…”

“…be positive measures where σ > 0 and τ > 0 are prescribed parameters. We define (Nielsen et al, 2017) a tri-parametric family of divergences as follows:…”

Synthesis, experimental andin silicostudies ofN-fluorenylmethoxycarbonyl-O-tert-butyl-N-methyltyrosine, coupled with CSD data: a survey of interactions in the crystal structures of Fmoc–amino acids

“…Density functional theory (DFT) calculations were used to determine the molecular structure of (1) in the gas phase and/ or in water solution. The initial conformations used in calculations were constructed using the GaussView graphical interface (Nielsen & Holder, 2009). We took the relative orientations of the bulky substituents of the carbamate group from the X-ray data of the crystal structure discussed earlier.…”

The Statistical Minkowski Distances: Closed-Form Formula for Gaussian Mixture Models

“…Informally speaking, a divergence 2 is a smooth distance 3 that allows one to define an information-geometric structure [2]. In other words, a divergence is a smooth premetric distance [9].Recently, the Cauchy-Schwarz divergence [18] has been generalized to Hölder divergences [39]. These Cauchy and Hölder distances D(p : q) are said to be projective because D(λp : λ ′ q) = D(p : q) for any λ, λ ′ > 0.…”

“…Recently, the Cauchy-Schwarz divergence [18] has been generalized to Hölder divergences [39]. These Cauchy and Hölder distances D(p : q) are said to be projective because D(λp : λ ′ q) = D(p : q) for any λ, λ ′ > 0.…”