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We consider the f-divergence, which takes the form $$ D_f(P \| Q) = \int_\Omega f\left(\frac{dP}{dQ}\right) dQ. $$ For example, when $f(t) = t \log t$, we obtain the KL-divergence.

My question is that, as a function of $P$, under what regularity conditions on $f$ and $Q$ is the f-divergence $D_f(P \| Q)$ (geodesically) $\mu$-smooth with respect to the Wasserstein metric $W_2$?

For any function $g(P)$, the (geodesic) $\mu$-smoothness of $g(P)$ takes the form $$ g(P') \leq g(P) + \langle\text{grad}\,g(P), \exp_P^{-1} (P') \rangle_P + \frac{\mu}{2}\cdot W_2^2(P, P') $$ or equivalently that the Hessian $\text{Hess}(g)$ is bounded from the above by $\mu$ for all $P$. Here, $\text{grad}$ and $\text{Hess}$ are the gradient and Hessian with respect to the $W_2$ metric, which is the geodesic distance in the space of probability distributions, and $\exp_P^{-1} (P')$ is the inverse of the exponential map.

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    $\begingroup$ Try searching bibliography using"McCann's displacement convexity" as a keyword, Robert McCann's paper [A convexity principle for interacting gases, 1997] is a good starting point. And perhaps also "$\lambda$-displacement convex" or "$\lambda$-geodesically convex" instead of $\mu$-smooth (the latter is not the usual terminology, at least not in my optimal transport community) $\endgroup$
    – leo monsaingeon
    Commented May 13, 2019 at 20:41
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    $\begingroup$ Just a heads-up: this is a really delicate topics, which gave rise in the last $\sim$10 years to the so-called Sturm-Lott-Villani theory of synthetic curvature. Roughly speaking, the geodesic convexity of your functional will arise from an interplay between some suitable structural/dimensional conditions on $f$ and the Ricci curvature of $\Omega$. $\endgroup$
    – leo monsaingeon
    Commented May 13, 2019 at 20:44

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