Sampling regime
Log-smooth + PI or LSI
Compare theorems by mathematical guarantee, rate, and source — not by raw crawler rows.
Theorem and frontier cases
best upper · Normalized SampleWiki statement
Implemented proximal sampler
Primary theorem audit pending — this is not presented as a verbatim paper theorem
Guarantee / accuracy
\[\lVert\mu_N-\pi\rVert_{\rm TV}\le\varepsilon\]
Complexity / rate
\[\widetilde O(\kappa\sqrt d\,\operatorname{polylog}(\sqrt{\operatorname{KL}_0}/\varepsilon^2))\]
lower unknown · Open problem
Matching PI/LSI oracle lower bound
Matching lower bound not currently known in the pinned literature trail
Matching result remains open.
upper · Exact source theorem
LMC
Theorem 4.2.6 (Vempala–Wibisono 2019; current Chewi numbering)
Theorem statement
\[\begin{gathered}C_{\mathrm{LSI}}(\pi)\le \alpha^{-1},\quad \nabla V\text{ is }\beta\text{-Lipschitz},\quad h\le (4\beta)^{-1},\\ \operatorname{KL}(\widehat\mu_{Nh}\Vert\pi)\le e^{-\alpha Nh}\operatorname{KL}(\widehat\mu_0\Vert\pi)+O\!\left(\frac{\beta^2dh}{\alpha}\right).\end{gathered}\]
upper · Normalized SampleWiki statement
LMC Rényi interpolation
Primary theorem audit pending — this is not presented as a verbatim paper theorem
Guarantee / accuracy
\[\sqrt{\mathcal R_q}\le\varepsilon\]
Complexity / rate
\[\widetilde O(\kappa^2dq\,\varepsilon^{-2}\log\mathcal R_2(\mu_0\Vert\pi))\]
upper · Normalized SampleWiki statement
ULMC warm-start construction
Primary theorem audit pending — this is not presented as a verbatim paper theorem
Guarantee / accuracy
\[\lVert\mu_N-\pi\rVert_{\rm TV}\le\varepsilon\]
Complexity / rate
\[\widetilde O(\kappa^{3/2}\sqrt d/\varepsilon)\]