Displacement convexity of generalized relative entropies

By Shin-ichi Ohta, Asuka Takatsu. 2010

49 citations recorded by the source corpus.

Abstract We investigate the m -relative entropy, which stems from the Bregman divergence, on weighted Riemannian and Finsler manifolds. We prove that the displacement K -convexity of the m -relative entropy is equivalent to the combination of the nonnegativity of the weighted Ricci curvature and the K -convexity of the weight function. We use this to show appropriate variants of the Talagrand, HWI and the logarithmic Sobolev inequalities, as well as the concentration of measures. We also prove that the gradient flow of the m -relative entropy produces a solution to the porous medium equation or the fast diffusion equation.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1f4616c4c69aac6ab5431ee0600837f0d479b677

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

(K, N)-Convexity and the Curvature-Dimension Condition for Negative N (2013, background): “Admitting N < 0 in RicN and CD(K , N ) may sound strange if one sticks to the image that N represents an upper bound of the dimension, however, its usefulness has already been recognized in the author’s work [33,34] with Takatsu (see also a related work [35] in the PDE theory).”
A warped product version of the Cheeger-Gromoll splitting theorem (2015, result): “In particular, this allows one to make sense of the CD ( K, N ), N < 0, condition on non-smooth spaces; also see the earlier works of Ohta-Takatsu [OT11, OT13].”
Poincaré and Brunn-Minkowski inequalities on the boundary of weighted Riemannian manifolds (2017, result): “[46, 72]); an exception is the work of Ohta and Takatsu [60, 61].”
Brascamp–Lieb-Type Inequalities on Weighted Riemannian Manifolds with Boundary (2013, result): “It seems that outside the Euclidean setting, this extension of the Curvature-Dimension condition to the range N ≤ 0 has not attracted much attention in the weighted-Riemannian and more general metric-measure space setting (cf. [44, 27]); an exception is the work of Ohta and Takatsu [40, 41].”
Beyond traditional Curvature-Dimension I: new model spaces for isoperimetric and concentration inequalities in negative dimension (2014, background): “Very recently, a growing interest in weighted manifolds with negative generalized dimension N < 0 has begun to emerge; see the works by Ohta–Takatsu [55,56], Ohta [54], Kolesnikov– Milman [31] and Klartag [30].”
SVGD as a kernelized Wasserstein gradient flow of the chi-squared divergence (2020, methodology): “The existence, uniqueness, and regularity of this ﬂow are studied in [OT11,OT13] and [AGS08, Theorem 11.2.1].”