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AbstractWe 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 Holder ordinary and reverse inequalities and present two novel classes of Holder 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 Holder distances are invariant to rescaling and thus do not require distributions to be normalized. Finally, we show how to compute statistical Holder 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 Holder divergences outperform the symmetric Cauchy-Schwarz divergence.
CitationNielsen F, Sun K, Marchand-Maillet S (2017) On Hölder Projective Divergences. Entropy 19: 122. Available: http://dx.doi.org/10.3390/e19030122.
SponsorsThe authors gratefully thank the referees for their comments. Ke Sun is funded by King Abdullah University of Science and Technology (KAUST).
Except where otherwise noted, this item's license is described as This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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How diking affects the longer-term structure and evolution of divergent plate boundariesTrippanera, Daniele; Acocella, Valerio; Ruch, Joel; Rivalta, Eleonora (2015-04)Recurrent diking episodes along divergent plate boundaries, as at Dabbahu (2005, Afar) or at Bardarbunga (2014, Iceland) , highlight the possibility to have m-wide opening in a short time (days to weeks). This suggests a prominent role of magma enhancing transient plate separations. However, the role of diking on a longer term (> 102 years) and its influence on the structure and the evolution of a divergent plate boundary is still poorly investigated. Here we use field surveys along the oceanic Icelandic and continental Ethiopian plate boundaries, along five eruptive fissures and four rift segments. Field observations have also been integrated with analogue and numerical models of dike emplacement to better understand the effect of dike emplacement at depth and at the surface. Our results show that the dike-fed eruptive fissures are systematically associated with graben structures formed by inward dipping normal faults having throws up to 10 m and commonly propagating downward. Moreover, rift segments (i.e. mature rift zones), despite any asymmetry and repetition, are characterized by the same features as the eruptive fissures, the only difference lying in the larger size (higher fault throws, up to 40 m, and wider deformation zones). Analogue and numerical models of dike intrusion confirm that all the structural features observed along the rift segments may be dike-induced; these features include downward propagating normal faults bordering graben structures, contraction at the base of the hanging walls of the faults and upward propagating faults. Simple calculations based on the deeper structure of the eroded rift segments in eastern and western Iceland also suggest that all the fault slip in the active rift segments may result from diking. These results suggest that the overall deformation pattern of eruptive fissures and rift segments may be explained only by dike emplacement. In a magmatic rift, the regional tectonic stress may rarely be high enough to be released through regional faulting, suggesting that regional tectonics has negligible direct impact compared to diking in shaping the studied plate boundaries on the longer-term.
Fault and graben growth along active magmatic divergent plate boundaries in Iceland and EthiopiaTrippanera, D.; Acocella, V.; Ruch, Joel; Abebe, B. (Wiley-Blackwell, 2015-10-08)Recent studies highlight the importance of annual-scale dike-induced rifting episodes in developing normal faults and graben along the active axis of magmatic divergent plate boundaries (MDPB). However, the longer-term (102-105 years) role of diking on the cumulative surface deformation and evolution of MDPB is not yet well understood. To better understand the longer-term normal faults and graben along the axis of MDPB, we analyze fissure swarms in Iceland and Ethiopia. We first focus on the simplest case of immature fissure swarms, with single dike-fed eruptive fissures; these consist of a <1 km wide graben bordered by normal faults with displacement up to a few meters, consistent with theoretical models and geodetic data. A similar structural pattern is found, with asymmetric and multiple graben, within wider mature fissure swarms, formed by several dike-fed eruptive fissures. We then consider the lateral termination of normal faults along these graben, to detect their upward or downward propagation. Most faults terminate as open fractures on flat surface, suggesting downward fault propagation; this is consistent with recent experiments showing dike-induced normal faults propagating downward from the surface. However, some normal faults also terminate as open fractures on monoclines, which resemble fault propagation folds; this suggests upward propagation of reactivated buried faults, promoted by diking. These results suggest that fault growth and graben development, as well as the longer-term evolution of the axis of MDPB, may be explained only through dike emplacement and that any amagmatic faulting is not necessary.