53 problems

Triangle inequality conjecture. The quantity $D_{\Delta}$ satisfies the triangle inequality.

Let $\mathbf{W}^{(n)}=\mathrm{W}_2^2(\hat\mu^{1,(n)},\hat\nu^{1,(n)})$ be the squared Wasserstein distance between two empirical samples of size $n$ from the standard normal distri…

Let $x_1>x_2>\cdots>x_N$ be the symmetric ground state particle locations satisfying … Let the empirical distribution of $\{x_n:n=1,\ldots,N\}$ be compared with the standard Gaussi…

Steinerberger's conjecture. There exists a constant $C$, depending only on $p$ and on the manifold, such that

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 $Y_1,\ldots,Y_n$ be independent identically distributed samples from a Gaussian distribution, let $\mathbf{y}=(Y_1,\ldots,Y_n)$, let $\bar{\mathbf{y}}$ be their sample mean, an…

Let $\bar\mu^{\theta^*}_s$ denote the law of the limiting system and let $\mu^{N,\theta^*}_s$ denote the empirical measure of the finite system with population size $N$. For any…

Let $X_1,X_2,\ldots$ be independent identically distributed random variables with mean $0$ and variance $\sigma^2$, let $S_n=X_1+\cdots+X_n$, and write $P_{S_n}$ for the law of…

Consider a nonlinear diffusion model on the sphere whose interaction kernel has an eigenfunction expansion … where the $Y_l$ are eigenfunctions of the Laplace–Beltrami operator, an…

Diagonal-coupling conjecture. The supremum

Bounded-activation extension conjecture. The main quantitative convergence result remains valid even without the bounded-activation Assumption.

Let $D\geq 2$. For the unit cube $[0,1]^D$ with its $\ell^{\infty}$ metric and an absolutely continuous measure whose density is bounded away from both $0$ and $\infty$, the paper…

Let $(\mathfrak{X},d,\mu)$ be a metric Polish probability space. The average-case bound refers to high-probability approximation of expectations of Lipschitz functions by empirical…

Let $M$ be a closed manifold of dimension $d$, let $V$ be the potential defining the invariant measure $mu$ for the diffusion generator, and let $p$ be the Wasserstein exponent. Fo…

Let $X_1,X_2,\ldots$ be independent random variables uniformly distributed on the unit cube $Q=(0,1)^d$, let $\mu_n=\sum_{i=1}^n\delta_{X_i}$, and define … Set … and … where…

Let the data domain be a finite metric space, and consider private density estimation under Wasserstein distance with instance-optimal guarantees. Logarithmic-factor necessity conj…

Let $\mu$ be a sub-Gaussian probability measure, and let $\mu_n$ be its empirical measure. For a smoothing parameter $\sigma>0$, write $\gamma_\sigma$ for the centered Gaussian mea…

Caracciolo–Lucià–Parisi–Sicuro conjecture. The renormalized expected cost has a finite limit:

Let $(M,\rho)$ be a connected complete compact weighted Riemannian manifold without boundary, equipped with a weighted probability measure $\mathrm{d}\mu=\mathrm{e}^V\mathrm{d}x$,…

Rio's conjecture. The optimal bound

Let $(I_n)$ be an increasing sequence of finite index sets with $|I_n|\to\infty$, and let $(X_i)_{i\in I}$ be centered random variables. Write $W_n$ for the normalized sum in the c…

Let $(I_n)$ be an increasing sequence of finite index sets with $|I_n|\to\infty$, and let $(X_i)_{i\in I}$ be centered random variables. Set … For probability measures $\nu$ and…

For $p\geq 1$, let $W_p$ denote the Wasserstein-$p$ distance. Under the moment and weak-dependence assumptions considered for the central limit theorem, the expected convergence ra…