61 problems

Let the graph size be $n$, and consider a random geometric graph in the sparse regime, where the edge probability vanishes with $n$. Geometry is conjectured to be lost at dimension…

Near-optimality conjecture. This bound is conjectured to be near-optimal when $\Sigma_W$ is unrestricted.

Mean-field critical behavior conjecture. This is conjectured to hold for $d>6$.

Refined asymptotic conjecture. Under further assumptions on the sequence $(\lambda(i))$, one can prove that

Mesoscopic multiplier estimate. There are absolute constants $C,c_0>0$ such that, in the width band above, if $d\ge\varepsilon^{-2}\log^C n$, then with probability at least $1-1/n$…

Let $\boldsymbol{X}\times N(0,\boldsymbol{I}_n)$ be a genuine sample, and let a fake sample have the form $\boldsymbol{X}+r\boldsymbol{t}(\boldsymbol{X})$, where the adversary choo…

Consider a sufficiently regular sequence of convex bodies $(K_n)_{n\geq 1}$ such that $\dim K_n=n$ for every $n$ and … as $n\to\infty$. For each $n$, let…

Let $g_1$ be a differentiable function, let $U$ be an orthogonal matrix, and define $g_2(\bm{y}):= g_1(U\bm{y})$ under the coordinate transformation $\bm{y}=U^T\bm{x}$. Let…

A convex set is in isotropic position when it has unit volume and its covariance matrix is a scalar multiple of the identity. Strong slicing conjecture. Among all unit-volume conve…

Proof-artifact conjecture. The restriction that $d/\lambda$ be sufficiently small can be dropped using a different proof technique. The conjecture asserts that the stated remainder…

Bubeck et al.'s geometry-loss conjecture. The geometry of a sparse RGG is lost once

Plan–Vershynin conjecture. A dependence of order $\delta^{-2}$ times the square of the Gaussian width should be necessary and sufficient for a random hyperplane tessellation on the…

Let $H_t(\mathcal{G})$ denote the entropy of the random geometric graph distribution at parameter $t$, and let $\bar{p}_{\max}$ be the corresponding edge-probability parameter at w…

KLS conjecture. The quantity $\Psi_n$ is bounded above by a universal constant, independent of the dimension $n$.

Let $T\subseteq\mathbb{R}^n$ be a set of adversarial tricks, let $X$ have the standard normal distribution in $\mathbb{R}^n$, and let the corrupted observation be $X+rt$, where…

Let $K$ be a convex body equipped with its uniform probability measure. Consider divisions of $K$ into two measurable pieces of equal mass, and measure the interface between the pi…

Let $d$ be fixed, and consider the LSS estimator $\hat{\pi}_{\mathrm{LSS}}$ under the assumptions of the fixed-dimensional model. Its error upper bound has the form … where…

Let $M\in\{\pm1\}^{n\times m}$ with $m\geq n$, and let $\operatorname{Res}(M)$ be the least number of entry-flips needed to produce from $M$ a matrix whose rank is strictly less th…

Exponential-scaling conjecture. The coefficient of variation scales exponentially in $\theta$. The conjecture concerns the dependence of fluctuations in the Fermat distance on the…

Let $n$ standard Gaussian random vectors in $\mathbb{R}^d$ be given, with $n,d\to\infty$ such that $n/d^2\to\alpha>0$. The fitting problem asks whether there exists a centered elli…

Let $P$ be the transition kernel of the geodesic random walk on the sphere for a constant target density, and let the contraction-rate estimates obtained from the coupling argument…

Let $\sigma_f(\boldsymbol{x},\boldsymbol{y})=F((\langle\boldsymbol{x},\boldsymbol{y}\rangle-\mu_{p,d,r})/(r\sqrt d))$, where $f$ is a $C^2$ $\alpha$-Lipschitz density of a zero-mea…

Let $\tau_{p,d}$ be chosen so that the spherical threshold connection…

Let $S_n^{(d)}$ be the random walk, let $\alpha$ and $L$ be as in the stable-domain-of-attraction condition … and let $\gamma\in(0,\infty)$. Assume that … Let $\zeta_\alpha$ be the…

For $p<\infty$, let $\mathsf{B}_p^n$ denote the unit ball of $\ell_p^n$, and consider its central hyperplane sections $\mathsf{B}_p^n\cap a^\perp$, where $a$ is a unit vector in…