Sampling is as easy as learning the score

Theorems 6-7 (critically damped Langevin). The CLD forward process is $d\bar X=\bar V\,dt$, $d\bar V=-(\bar X+2\bar V)\,dt+2\,dB$, and its reverse needs only the velocity score $\nabla_v\ln\boldsymbol q_t$. Under the analogous assumptions Theorem 6 gives the bound of Theorem 2, now on $\mathrm{TV}(p_T,q\otimes\gamma^d)$ and with first term $\sqrt{\mathrm{KL}(q\|\gamma^d)+\mathrm{FI}(q\|\gamma^d)}\,e^{-cT}$. Theorem 7 shows the technique cannot do better: for $q=\gamma^d$, exact scores and $h\le1/10$, the path-space divergence satisfies $\mathrm{KL}(\boldsymbol Q^\leftarrow_T\|\boldsymbol P_T)\ge dhT$, so the Girsanov route needs $h\lesssim1/d$, the same dimension dependence as DDPM.

The error splits into a part from starting at $\gamma^d$ instead of $q_T$ and a part from following the reverse path, both measured in information divergences. Compare the algorithm with the same algorithm started at $q_T$, whose path law is $P^{q_T}_T$, and that one with the true reverse path law $Q^\leftarrow_T$. Data processing (the same Markov kernel acts on both starts, and marginals are images of paths) and Pinsker give $$\mathrm{TV}(p_T,q)\le\mathrm{TV}(\gamma^d,q_T)+\mathrm{TV}(P^{q_T}_T,Q^\leftarrow_T)\le\sqrt{\mathrm{KL}(q\|\gamma^d)}\,e^{-T}+\sqrt{\mathrm{KL}(Q^\leftarrow_T\|P^{q_T}_T)},$$ where the first bound is the exponential decay of KL along OU from the log-Sobolev inequality of $\gamma^d$. Information divergences are used because the reverse dynamics can be expansive: for $q=N(0,\sigma^2I)$ the reverse drift near the end is $(1-2/\sigma^2)x$, which pushes trajectories apart when $\sigma^2\gt2$, so a $W_2$ coupling would need log-concavity or dissipativity and even then the error bound grows exponentially in time, while data processing in TV or KL holds for any dynamics.

Girsanov turns the tracking error into the score error along the true process. The two path laws share the noise $\sqrt2\,dB$ and the start $q_T$, so Girsanov's formula involves only the drift gap $b_t=\sqrt2\{s_{T-kh}(X_{kh})-\nabla\ln q_{T-t}(X_t)\}$, and since the stochastic integral has mean zero under $Q^\leftarrow_T$, $$\mathrm{KL}(Q^\leftarrow_T\,\|\,P^{q_T}_T)\le\sum_{k=0}^{N-1}\int_{kh}^{(k+1)h}\mathbb E_{Q^\leftarrow_T}\big\|s_{T-kh}(X_{kh})-\nabla\ln q_{T-t}(X_t)\big\|^2\,dt .$$ The direction of the divergence matters here. The expectation is under the true reverse process, whose time-$t$ marginal is $q_{T-t}$, so at grid points the score error is literally (A3), and the error under the algorithm's own law never has to be bounded. The earlier interpolation method differentiates the divergence along the algorithm, so it needs the score error under the algorithm's marginals, which forced a log-Sobolev assumption.

Each step contributes $\varepsilon_{\rm score}^2+L^2dh+L^2m_2^2h^2$, and the dominant term is the Brownian movement over one step, of size $dh$, times $L^2$ from the Lipschitz score. On $[kh,(k+1)h]$ split the integrand into estimation error, the change of the score in time, and the change of its argument: $$\mathbb E\|s_{T-kh}(X_{kh})-\nabla\ln q_{T-t}(X_t)\|^2\lesssim\varepsilon_{\rm score}^2+\mathbb E\Big\|\nabla\ln\frac{q_{T-kh}}{q_{T-t}}(X_{kh})\Big\|^2+L^2\,\mathbb E\|X_{kh}-X_t\|^2 .$$ The first term is (A3), and the last is $L^2$ times the path's squared movement over a time $\delta=t-kh\le h$, which is $\lesssim\delta^2m_2^2+\delta d$. The middle term compares two forward marginals $\delta$ apart: $q_{T-kh}$ is $q_{T-t}$ contracted by $e^{-\delta}$, which contributes an $L\delta\|\theta\|$ term, and smoothed by $N(0,(1-e^{-2\delta})I)$. The score perturbation lemma handles the smoothing by writing the change of score as the gap between $\nabla H$ at a point and its average over a small Gaussian tilt there: with $q=e^{-H}$ and $q^\theta\propto q\cdot N(\theta,M)$, $$\nabla\ln\frac{q*N(0,M)}{q}(\theta)=\nabla H(\theta)-\mathbb E_{q^\theta}\nabla H .$$ For $L\le1/(2\|M\|)$ the tilt is strongly log-concave, so it concentrates within $\sqrt{\|M\|d}$ of a mode that lies within $2\|M\|\|\nabla H(\theta)\|$ of $\theta$, and with $\|M\|=O(h)$ the middle term is $\lesssim L^2dh+L^2h^2(\|X_{kh}\|^2+\|\nabla\ln q_{T-t}(X_{kh})\|^2)$. Bounding the moments ($\mathbb E\|\nabla\ln q_t\|^2\le Ld$ by integration by parts) and summing gives $\mathrm{KL}(Q^\leftarrow_T\|P^{q_T}_T)\lesssim(\varepsilon_{\rm score}^2+L^2dh+L^2m_2^2h^2)T$, and Step 1 gives Theorem 2. The $L^2dh$ term is the $dh$ of Brownian movement over one step passed through the Lipschitz score, which is why $h\asymp\varepsilon^2/(L^2d)$ and $N$ is linear in $d$.

For bounded data, stop early and treat the slightly noised data as the target. Without a smooth density, (A1) fails at $t=0$ and $\mathrm{KL}(q\|\gamma^d)$ may be infinite, but the OU marginal $q_t$ at a small time satisfies every hypothesis of Theorem 2, though with a Lipschitz constant that grows like $R^2/t^2$. The explicit OU solution gives $W_2^2(q_t,q)\lesssim R^2t^2+dt$, at most $\varepsilon^2$ for the stated $t$; Otto-Villani short-time regularization makes the start-up term finite, $\mathrm{KL}(q_t\|\gamma^d)\lesssim W_2^2(q,\gamma^d)/t\lesssim\sqrt d(R\vee\sqrt d)^3/\varepsilon^2$; and the second-order Tweedie identity bounds the Hessian through the posterior covariance of $\bar X_0$ given $\bar X_t$, which is at most $R^2I$: $$\frac{1}{1-e^{-2t}}I-\frac{e^{-2t}R^2}{(1-e^{-2t})^2}I\ \preceq\ -\nabla^2\ln q_t\ \preceq\ \frac{1}{1-e^{-2t}}I,$$ so every $q_{t'}$, $t'\ge t$, has $L\lesssim R^2/t^2\lesssim dR^2(R\vee\sqrt d)^2/\varepsilon^4$. Theorem 2 with target $q_t$ gives Corollary 3, and $N\propto L^2d$ is where $d^3R^4(R\vee\sqrt d)^4/\varepsilon^8$ comes from. For $W_2$, project onto a ball: there $W_2$ is at most the radius times $\sqrt{\mathrm{TV}}$ plus an exponentially small tail term, and the projection is 1-Lipschitz.

CLD gains nothing, because its score is taken in the velocity, which moves like Brownian motion. Theorem 6 repeats Steps 2-3 with $\nabla_v\ln\boldsymbol q_t$, replacing OU's KL decay by entropic hypocoercivity. For Theorem 7 take $q=\gamma^d$: the joint law stays $\gamma^{2d}$, the exact velocity score is $-v$, Novikov holds directly, and Girsanov is an identity whose integrand is the velocity's own increment, $$\mathrm{KL}(\boldsymbol Q^\leftarrow_T\|\boldsymbol P_T)=2\sum_k\int_{kh}^{(k+1)h}\mathbb E\|V_{kh}-V_t\|^2dt\ \ge\ 2\sum_k\int_{kh}^{(k+1)h}d\,(t-kh)\,dt=dhT,$$ using $\mathbb E\|V_{kh}-V_t\|^2\ge2d(t-kh)-10d(t-kh)^2\ge d(t-kh)$ for $h\le1/10$. The velocity carries the Brownian increment. Underdamped Langevin gets its $\sqrt d$ speedup in log-concave sampling because its gradient depends only on the smooth position, whose increment is $O(dh^2)$; the reverse CLD drift depends on $V$, so the speedup is lost. This is a lower bound for the technique only; the authors conjecture, without proof, that CLD has no better dimension dependence in general.