283 problems

Let $(athcal{M},\mathcal{W}^{a,b}_p)$ be the Wasserstein metric space of nonnegative measures on a finite Markov chain with state space $\mathcal{X}$, and let $t\mapsto\mu_t$ be a…

Let $n\ge1$ and $k\ge2$. Let $\rho:\mathbb R\to\mathbb R_+$ be non-increasing, and let $f_1,\ldots,f_k:\mathbb R^n\to\mathbb R_+$ be even integrable functions. For … assume … for a…

De Palma–Trevisan conjecture. The divergences defined this way are genuine metrics on quantum state spaces. The conjecture has recently been justified under certain additional assu…

Persistent homology equivalence conjecture. The infinity-Vietoris–Rips metric thickenings should have the same persistent homology as the classical Vietoris–Rips simplicial complex…

Klartag's conjecture. The localisation method should generalise to the multidimensional setting by employing optimal transport of vector measures.

Let $d$ be a positive integer, let $(\mathbb{R}^d,g,\mu)$ satisfy $\operatorname{CD}(\rho,\infty)$ for some $\rho>0$, and define the centered Gaussian probability measure … where…

Villani's conjecture. If the squared-distance cost satisfies the MTW condition, then all injectivity domains of $M$ are convex.

Let $\Omega$ be a convex and bounded domain in $\mathbb{R}^d$ with smooth boundary, and let $f^{+},f^{-}:\Omega\rightarrow\mathbb{R}_{\geq 0}$ satisfy … For a positive initial datu…

The setting is a hypergraph constructed by uniformly sampling a Riemannian manifold in the natural way, with coarse scalar curvature and the manifold's scalar curvature compared as…

Wasserstein distance superiority conjecture. For a very large training set, the Wasserstein distance will have better performance than each of the three tested distances.

Let $F_0$ be a probability density on $\mathbb{R}^d$, and let $F_1$ be a minimizer of the unconstrained variational problem … Here $S$ is the entropy functional and $W_2$ is the…

Adaptive field codes represent smooth transport fields using only the cells with nonzero perturbation, while certified sampler protocols may also require residual exact-marginal co…

Let $m$ be a positive integer, let $1<p,q\le 2$, and define the rectangular power cost … For each $\varepsilon\in\{-1,1\}$, define the finite diamond permutation of…

Asymptotic linear-form conjecture. Under the setting of Theorem InfluenceQuantiles-main,

Density conjecture for compositions of optimal maps. The set of compositions of $k=4$ optimal maps is dense in the set of orientation-preserving diffeomorphisms, whereas the analog…

Let $C^Q$ be the projector onto the antisymmetric subspace, … For density matrices $\rho_1,\rho_2$, let $W_{C,2}$ denote the quantum $2$-Wasserstein semidistance associated with a…

Friedland et al.'s conjecture. For every dimension $n$, $W_{C^Q,2}$ satisfies the triangle inequality for all density matrices, and hence is a true distance. The conjecture has bee…

For an $n \times n$ distance matrix $(E_{ij})$, let … where $|i \wedge j\rangle:= \tfrac{1}{\sqrt{2}}(|ij\rangle-|ji\rangle)$, and let $W_{C_E,p}$ be the associated quantum $p$-Was…

Let $n\ge1$ and $k\ge2$, and let $K_1,\ldots,K_k\subset\mathbb R^n$ be origin-symmetric convex bodies. Kalantzopoulos–Saroglou's conjecture. If … for all $x_i\in K_i$,…

Let $(\mathcal{P}_2,\mathcal{W}_2)$ denote the Wasserstein space of probability measures with finite second moment, and let $\tilde g_{e,2J}$ be the cost metric considered in the p…

Let $k,l\ge 2$ and let $\pi\in\mathcal{P}([0,1]^k\times[0,1]^l)$ be a multivariate distribution. Multivariate adapted copula rate conjecture. Consistency should still hold, while t…

Let $k\ge1$. On a regular problem with an equispaced dense design, consider a degree-$k$ temporal local-polynomial forecaster on the tangent bundle, obtained by geodesic or barycen…

Let $M$ be the underlying manifold and let the Wasserstein–Ebin metric be the metric on the space of Riemannian metrics studied in the article. Its induced geodesic distance is the…

Single-sign-switch conjecture. The function $\iota$ switches sign once on the positive part of $I$.

Let $\mathcal{C}:[0,1]\times[0,1]\to\mathbb{R}$ be twice continuously differentiable up to the boundary, symmetric in its two variables, zero on the diagonal, and satisfy…