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leo monsaingeon
leo monsaingeon
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Let $\mathcal{P}({\mathbb{R}^d})$ denote the set of Borel probability measures on $\mathbb{R}^d$, and let $\pi \in \mathcal{P} (\mathbb{R}^d)$. It is known that the sets $\{ \mu \in \mathbb{R}^d : KL( \mu \mid \pi ) \leq C\}$$\{ \mu \in \mathcal{P}(\mathbb{R}^d) :\, KL( \mu \mid \pi ) \leq C\}$ are compact with respect to the weak convergence of probability measures, as well as w.r.t. setwise convergence. What about with respect to the 2-Wasserstein metric? (If one replaces $\mathcal{P}({\mathbb{R}^d})$ with $\mathcal{P}_2({\mathbb{R}^d})$.)

Let $\mathcal{P}({\mathbb{R}^d})$ denote the set of Borel probability measures on $\mathbb{R}^d$, and let $\pi \in \mathcal{P} (\mathbb{R}^d)$. It is known that the sets $\{ \mu \in \mathbb{R}^d : KL( \mu \mid \pi ) \leq C\}$ are compact with respect to the weak convergence of probability measures, as well as w.r.t. setwise convergence. What about with respect to the 2-Wasserstein metric? (If one replaces $\mathcal{P}({\mathbb{R}^d})$ with $\mathcal{P}_2({\mathbb{R}^d})$.)

Let $\mathcal{P}({\mathbb{R}^d})$ denote the set of Borel probability measures on $\mathbb{R}^d$, and let $\pi \in \mathcal{P} (\mathbb{R}^d)$. It is known that the sets $\{ \mu \in \mathcal{P}(\mathbb{R}^d) :\, KL( \mu \mid \pi ) \leq C\}$ are compact with respect to the weak convergence of probability measures, as well as w.r.t. setwise convergence. What about with respect to the 2-Wasserstein metric? (If one replaces $\mathcal{P}({\mathbb{R}^d})$ with $\mathcal{P}_2({\mathbb{R}^d})$.)

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pseudocydonia
pseudocydonia
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Let $\mathcal{P}({\mathbb{R}^d})$ denote the set of Borel probability measures on $\mathbb{R}^d$, and let $\pi \in \mathbb{R}^d$$\pi \in \mathcal{P} (\mathbb{R}^d)$. It is known that the sets $\{ \mu \in \mathbb{R}^d : KL( \mu \mid \pi ) \leq C\}$ are compact with respect to the weak convergence of probability measures, as well as w.r.t. setwise convergence. What about with respect to the 2-Wasserstein metric? (If one replaces $\mathcal{P}({\mathbb{R}^d})$ with $\mathcal{P}_2({\mathbb{R}^d})$.)

Let $\mathcal{P}({\mathbb{R}^d})$ denote the set of Borel probability measures on $\mathbb{R}^d$, and let $\pi \in \mathbb{R}^d$. It is known that the sets $\{ \mu \in \mathbb{R}^d : KL( \mu \mid \pi ) \leq C\}$ are compact with respect to the weak convergence of probability measures, as well as w.r.t. setwise convergence. What about with respect to the 2-Wasserstein metric? (If one replaces $\mathcal{P}({\mathbb{R}^d})$ with $\mathcal{P}_2({\mathbb{R}^d})$.)

Let $\mathcal{P}({\mathbb{R}^d})$ denote the set of Borel probability measures on $\mathbb{R}^d$, and let $\pi \in \mathcal{P} (\mathbb{R}^d)$. It is known that the sets $\{ \mu \in \mathbb{R}^d : KL( \mu \mid \pi ) \leq C\}$ are compact with respect to the weak convergence of probability measures, as well as w.r.t. setwise convergence. What about with respect to the 2-Wasserstein metric? (If one replaces $\mathcal{P}({\mathbb{R}^d})$ with $\mathcal{P}_2({\mathbb{R}^d})$.)

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pseudocydonia
pseudocydonia
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Wasserstein compactness of sublevel sets of relative entropy

Let $\mathcal{P}({\mathbb{R}^d})$ denote the set of Borel probability measures on $\mathbb{R}^d$, and let $\pi \in \mathbb{R}^d$. It is known that the sets $\{ \mu \in \mathbb{R}^d : KL( \mu \mid \pi ) \leq C\}$ are compact with respect to the weak convergence of probability measures, as well as w.r.t. setwise convergence. What about with respect to the 2-Wasserstein metric? (If one replaces $\mathcal{P}({\mathbb{R}^d})$ with $\mathcal{P}_2({\mathbb{R}^d})$.)