47 problems

Davies–Perkins conjecture. The down-up walk mixes in polynomial time whenever

Let $G$ be a graph of maximum degree $Delta$, let $delta\in(0,1)$, and consider the paper's particle-system algorithm for sampling independent sets with prescribed marginals. Exten…

Dimension-square mixing conjecture. The mixing time could be

Bounded relaxation-levels conjecture. We conjecture that $L$ is bounded independently of $n$, the number of vertices in the input graph. This would yield a linear-time parallelizab…

Efficient-sampling conjecture. A Poincaré inequality together with bounded outer isoperimetry $$

Let $g$ be the function used to impose oscillatory dynamics in the point-process sampler, and let mixing time and asymptotic variance be efficiency metrics. Oscillatory-dynamics ef…

Consider the point-process sampler and the birth-death sampler for a target distribution on a discrete state space, and let $\textsc{ess}$ denote the effective sample size. Point-p…

Let $M_{Q,t}/D/\infty$ denote the potentially coupled point-process sampler with deterministic service times, let $M_{Q,t}/M_t/\infty$ denote the corresponding potentially coupled…

Consider the spiked Wigner inference problem with signal-to-noise parameter $\beta_{\bM}$ and threshold $\beta_{\bM}$ as used in the paper. Let $\beta_{\bM}$ denote the scaled post…

Balanced-schedule convergence conjecture. When the schedule is balanced, matching upper and lower bounds of order $1/T$ should be attainable.

KLS conjecture. The Poincaré constant satisfies $C_{\mathsf{PI}}(\pi)=\mathcal{O}(1)$, uniformly over all dimensions and isotropic logconcave probability measures.

Let $X_u$ be the CREM field on the binary tree, with $u e0$ denoting a vertex and $|u|$ its depth. Suppose that an algorithm has access only to the leaf values $Xu$ for $|u|=N$. **…

Let $R_{\epsilon}^{\phi_s}(D)$ denote the rate-distortion function at average distortion constraint $D$ when the initial sampling phase is $\phi_s$, and let $R_{\epsilon}^{0}(D)$ d…

Computational hardness conjecture. Beyond the shattering phase transition, sampling from the Gibbs measure is fundamentally hard, not merely slow for Langevin dynamics.

Proportional error-bound conjecture. For other sampling strategies, such as ergodic sampling, similar arguments to those used for the proportional error bound should apply if the e…

Let $D_d\subset\mathbb{R}^d$ be a domain, let $\mathcal{P}_n$ denote a set of $n$ sampling points, and consider the integration or approximation error … Here $W_p^s(D_d)$ is a Sobo…

Let $\kappa=M/m$ denote the condition number, and let $\varepsilon>0$ be the target accuracy in Wasserstein distance. Kinetic Langevin complexity conjecture. The number of iteratio…

Let $\kappa=M/m$ be the condition number associated with a strongly log-concave target, and let kinetic Langevin Monte Carlo use a non-synchronous coupling. Improved kinetic Langev…

Consider the MCMC setting for sampling from a distribution associated with a possibly non-concave function $f$, where the available complexity bounds are measured in gradient or fu…

Density-based sampling norm-dependence conjecture. The dependence on $\|f\|$ in the polynomial-time sampling rate is similar to the dependence in this bound. The conjecture concern…

The posterior distribution $\rho_{\rm post}(\theta)$ is strongly log-concave, and the convergence-rate constants for the Gaussian approximate gradient flow and Gaussian approximate…

Let $\rho_t$ denote the solution of the Langevin Fokker–Planck equation with initial density $\rho_0$, and let the target density $\pi$ satisfy the log-Sobolev inequality (LSI). As…

Let critically damped Langevin diffusion (CLD) denote the diffusion used as the forward process in a score-based generative model (SGM), and let DDPM denote the original denoising…

El Alaoui–Montanari–Selke sampling conjecture. The algorithm successfully samples from the Sherrington–Kirkpatrick Gibbs measure for every $\lambda<1$.

El Alaoui–Montanari–Selke conjecture. For $\lambda<1$, sampling from $\mu_{\boldsymbol W}$ is possible in polynomial time; in particular, their algorithm achieves this.