cs.LGApr 12, 2026

Rethinking the Diffusion Model from a Langevin Perspective

Authors: Candi Zheng, Yuan Lan

Organizations: Department of Mathematics, The Hong Kong University of Science and Technology

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Abstract

Diffusion models are often introduced from multiple perspectives, such as VAEs, score matching, or flow matching, accompanied by dense and technically demanding mathematics that can be difficult for beginners to grasp. One classic question is: how does the reverse process invert the forward process to generate data from pure noise? This article systematically organizes the diffusion model from a fresh Langevin perspective, offering a simpler, clearer, and more intuitive answer. We also address the following questions: how can ODE-based and SDE-based diffusion models be unified under a single framework? Why are diffusion models theoretically superior to ordinary VAEs? Why is flow matching not fundamentally simpler than denoising or score matching, but equivalent under maximum-likelihood? We demonstrate that the Langevin perspective offers clear and straightforward answers to these questions, bridging existing interpretations of diffusion models, showing how different formulations can be converted into one another within a common framework, and offering pedagogical value for both learners and experienced researchers seeking deeper intuition.

Figures & tables

Figure 1
Figure 1: Langevin dynamics acts as an identity operation on p(x)p(\mathbf{x}) : starting from a sample x∼p(x)\mathbf{x}\sim p(\mathbf{x}) , it produces a new sample x′\mathbf{x}^{\prime} from the same distribution.
Model TypeNoise-level parameterForward processForward SDE
VPαt=e−t\alpha_{t}=e^{-t}xt=αt x0+1−αt ϵx_{t}=\sqrt{\alpha_{t}}\,x_{0}+\sqrt{1-\alpha_{t}}\,\bm{\epsilon}dxt=−12xt dt+dWtdx_{t}=-\tfrac{1}{2}x_{t}\,dt+dW_{t}
VE-Karrasσ\sigmazσ=z0+σ ϵz_{\sigma}=z_{0}+\sigma\,\bm{\epsilon}dzσ=2σ dWσdz_{\sigma}=\sqrt{2\sigma}\,dW_{\sigma}
Rectified flowssrs=(1−s) r0+s ϵr_{s}=(1-s)\,r_{0}+s\,\bm{\epsilon}drs=−rs1−s ds+2s1−s dWsdr_{s}=-\frac{r_{s}}{1-s}\,ds+\sqrt{\frac{2s}{1-s}}\,dW_{s}
Table 1: Forward processes across model parameterizations.
Figure 2: Overview of forward processes across VP, VE-Karras, and rectified-flow parameterizations (exported from the original interactive visualization https://iclr-blogposts.github.io/2026/blog/2026/rethinking-diffusion-langevin/ ).
Given parameterizationEquivalent VPEquivalent VE-KarrasEquivalent Rectified-flow
VP (xt,αt)(x_{t},\alpha_{t})/zσ=xtαtz_{\sigma}=\dfrac{x_{t}}{\sqrt{\alpha_{t}}} [0.25em] σ=1−αtαt\sigma=\sqrt{\dfrac{1-\alpha_{t}}{\alpha_{t}}}rs=xtαt+1−αtr_{s}=\dfrac{x_{t}}{\sqrt{\alpha_{t}}+\sqrt{1-\alpha_{t}}} [0.25em] s=1−αtαt+1−αts=\dfrac{\sqrt{1-\alpha_{t}}}{\sqrt{\alpha_{t}}+\sqrt{1-\alpha_{t}}}
VE-Karras (zσ,σ)(z_{\sigma},\sigma)xt=zσ1+σ2x_{t}=\dfrac{z_{\sigma}}{\sqrt{1+\sigma^{2}}} [0.25em] αt=11+σ2\alpha_{t}=\dfrac{1}{1+\sigma^{2}}/rs=zσ1+σr_{s}=\dfrac{z_{\sigma}}{1+\sigma} [0.25em] s=σ1+σs=\dfrac{\sigma}{1+\sigma}
Rectified flow (rs,s)(r_{s},s)xt=rs(1−s)2+s2x_{t}=\dfrac{r_{s}}{\sqrt{(1-s)^{2}+s^{2}}} [0.25em] αt=(1−s)2(1−s)2+s2\alpha_{t}=\dfrac{(1-s)^{2}}{(1-s)^{2}+s^{2}}zσ=rs1−sz_{\sigma}=\dfrac{r_{s}}{1-s} [0.25em] σ=s1−s\sigma=\dfrac{s}{1-s}/
Table 2: Conversion between forward-process variable parameterizations.
Figure 3: A forward diffusion step with step size Δt\Delta t adds Gaussian noise to data, pushing samples closer to a Gaussian distribution.
Model TypeLangevin dynamicsReverse SplitForward Split
VP-SDEdx=sx dτ+2 dWτdx=\mathbf{s}_{x}\,d\tau+\sqrt{2}\,dW_{\tau}dx=[12x+sx]dτ+dWτdx=\left[\frac{1}{2}x+\mathbf{s}_{x}\right]d\tau+dW_{\tau}dx=−12x dτ+dWτdx=-\tfrac{1}{2}x\,d\tau+dW_{\tau}
VP-ODEdx=12sx dτ+dWτdx=\frac{1}{2}\mathbf{s}_{x}\,d\tau+dW_{\tau}dx=12(x+sx)dτdx=\frac{1}{2}\left(x+\mathbf{s}_{x}\right)d\taudx=−12x dτ+dWτdx=-\tfrac{1}{2}x\,d\tau+dW_{\tau}
VE-Karrasdz=τ sz dτ+2τ dWτdz=\tau\,\mathbf{s}_{z}\,d\tau+\sqrt{2\tau}\,dW_{\tau}dz=τ sz dτdz=\tau\,\mathbf{s}_{z}\,d\taudz=2τ dWτdz=\sqrt{2\tau}\,dW_{\tau}
Rectified flowdr=τ1+τsr dτ+2τ1+τ dWτdr=\frac{\tau}{1+\tau}\mathbf{s}_{r}\,d\tau+\sqrt{\frac{2\tau}{1+\tau}}\,dW_{\tau}dr=τ sr+r1−τdτdr=\frac{\tau\,\mathbf{s}_{r}+r}{1-\tau}d\taudr=−r1−τ dτ+2τ1−τ dWτdr=-\frac{r}{1-\tau}\,d\tau+\sqrt{\frac{2\tau}{1-\tau}}\,dW_{\tau}
Table 3: Langevin split of different model types.
Figure 4: The forward and reverse diffusion processes compose to reproduce Langevin dynamics.
Model TypeReverse ProcessRelation to ScoreReverse TimeReverse time domain
VP-SDEdxt′=[12xt′+s(xt′,T−t′)]dt′+dWt′d\mathbf{x}_{t^{\prime}}=\left[\frac{1}{2}\mathbf{x}_{t^{\prime}}+\mathbf{s}(\mathbf{x}_{t^{\prime}},T-t^{\prime})\right]dt^{\prime}+d\mathbf{W}_{t^{\prime}}s(x,t)=sx(x,t)\mathbf{s}(\mathbf{x},t)=\mathbf{s}_{x}(\mathbf{x},t)t′=T−tt^{\prime}=T-tt′∈[0,T]t^{\prime}\in[0,T]
VP-ODEdxt′=12[xt′+s(xt′,T−t′)]dt′d\mathbf{x}_{t^{\prime}}=\frac{1}{2}\left[\mathbf{x}_{t^{\prime}}+\mathbf{s}(\mathbf{x}_{t^{\prime}},T-t^{\prime})\right]dt^{\prime}s(x,t)=sx(x,t)\mathbf{s}(\mathbf{x},t)=\mathbf{s}_{x}(\mathbf{x},t)t′=T−tt^{\prime}=T-tt′∈[0,T]t^{\prime}\in[0,T]
VE-Karrasdzσ′=−ϵ(zσ′,Σ−σ′) dσ′d\mathbf{z}_{\sigma^{\prime}}=-\bm{\epsilon}(\mathbf{z}_{\sigma^{\prime}},\Sigma-\sigma^{\prime})\,d\sigma^{\prime}ϵ(z,σ)=−σsz(z,σ)\bm{\epsilon}(\mathbf{z},\sigma)=-\sigma\mathbf{s}_{z}(\mathbf{z},\sigma)σ′=Σ−σ\sigma^{\prime}=\Sigma-\sigmaσ′∈[0,Σ]\sigma^{\prime}\in[0,\Sigma]
Rectified flowdrs′=−v(rs′,1−s′) ds′d\mathbf{r}_{s^{\prime}}=-\mathbf{v}(\mathbf{r}_{s^{\prime}},1-s^{\prime})\,ds^{\prime}v(r,s)=−s sr(r,s)+r1−s\mathbf{v}(\mathbf{r},s)=-\frac{s\,\mathbf{s}_{r}(\mathbf{r},s)+\mathbf{r}}{1-s}s′=1−ss^{\prime}=1-ss′∈[0,1]s^{\prime}\in[0,1]
Table 4: Reverse diffusion processes across model types.
Given predictionEquivalent VP score sx\mathbf{s}_{x}Equivalent VE noise ϵ\bm{\epsilon}Equivalent RF velocity v\mathbf{v}
VP score sx(xt,αt)\mathbf{s}_{x}(x_{t},\alpha_{t})/ϵ(zσ,σ)=−1−αt sx(xt,αt)\bm{\epsilon}(z_{\sigma},\sigma)=-\sqrt{1-\alpha_{t}}\,\mathbf{s}_{x}(x_{t},\alpha_{t})v(rs,s)=−xtαt−1−αt+αt(1−αt)αt sx(xt,αt)\mathbf{v}(r_{s},s)=-\frac{x_{t}}{\sqrt{\alpha_{t}}}-\frac{1-\alpha_{t}+\sqrt{\alpha_{t}(1-\alpha_{t})}}{\sqrt{\alpha_{t}}}\,\mathbf{s}_{x}(x_{t},\alpha_{t})
VE noise ϵ(zσ,σ)\bm{\epsilon}(z_{\sigma},\sigma)sx(xt,αt)=−1+σ2σ ϵ(zσ,σ)\mathbf{s}_{x}(x_{t},\alpha_{t})=-\frac{\sqrt{1+\sigma^{2}}}{\sigma}\,\bm{\epsilon}(z_{\sigma},\sigma)/v(rs,s)=(1+σ)ϵ(zσ,σ)−zσ\mathbf{v}(r_{s},s)=(1+\sigma)\bm{\epsilon}(z_{\sigma},\sigma)-z_{\sigma}
RF velocity v(rs,s)\mathbf{v}(r_{s},s)sx(xt,αt)=−(1−s)2+s2s(rs+(1−s)v(rs,s))\mathbf{s}_{x}(x_{t},\alpha_{t})=-\frac{\sqrt{(1-s)^{2}+s^{2}}}{s}\left(r_{s}+(1-s)\mathbf{v}(r_{s},s)\right)ϵ(zσ,σ)=rs+(1−s)v(rs,s)\bm{\epsilon}(z_{\sigma},\sigma)=r_{s}+(1-s)\mathbf{v}(r_{s},s)/
Table 5: Conversion between model prediction.
Figure 5: Reverse trajectories under different parameterizations (exported from the original interactive visualization https://iclr-blogposts.github.io/2026/blog/2026/rethinking-diffusion-langevin/ ).
Figure 6: Part of a forward–reverse diffusion cycle: the last two steps of the forward process (green arrows, increasing tt ) followed by the first two steps of the reverse process (blue arrows, increasing t′t^{\prime} while decreasing tt ).
Figure 7: Each horizontal row shows a Langevin dynamics step that maps a forward sample xt\mathbf{x}_{t} to a new reverse sample x(T−t)′\mathbf{x}_{(T-t)^{\prime}} from the same probability density.
Model TypeNoise-state relationNetwork outputsθ\mathbf{s}_{\theta} w.r.t. NN∇log⁡p(xt∣x0)\nabla\log p(x_{t}\mid x_{0})Loss LtL_{t}
VPxt=αt x0+1−αt ϵx_{t}=\sqrt{\alpha_{t}}\,x_{0}+\sqrt{1-\alpha_{t}}\,\bm{\epsilon}sθ(xt,t)\mathbf{s}_{\theta}(x_{t},t)sθ(xt,t)\mathbf{s}_{\theta}(x_{t},t)−ϵ1−αt-\frac{\bm{\epsilon}}{\sqrt{1-\alpha_{t}}}12Ex0∼p0Ext∼pt(⋅∣x0)∥−ϵ1−αt−sθ(xt,t)∥2\frac{1}{2}\mathbb{E}_{\mathbf{x}_{0}\sim p_{0}}\mathbb{E}_{\mathbf{x}_{t}\sim p_{t}(\cdot\mid\mathbf{x}_{0})}\left\|-\frac{\bm{\epsilon}}{\sqrt{1-\alpha_{t}}}-\mathbf{s}_{\theta}(x_{t},t)\right\|^{2}
VE-Karraszσ=z0+σ ϵz_{\sigma}=z_{0}+\sigma\,\bm{\epsilon}ϵθ(zσ,σ)\bm{\epsilon}_{\theta}(z_{\sigma},\sigma)−ϵθ(zσ,σ)σ-\frac{\bm{\epsilon}_{\theta}(z_{\sigma},\sigma)}{\sigma}−ϵσ-\frac{\bm{\epsilon}}{\sigma}1σEz0∼p0Ezσ∼pσ(⋅∣z0)∥ϵθ(zσ,σ)−ϵ∥2\frac{1}{\sigma}\mathbb{E}_{\mathbf{z}_{0}\sim p_{0}}\mathbb{E}_{\mathbf{z}_{\sigma}\sim p_{\sigma}(\cdot\mid\mathbf{z}_{0})}\left\|\bm{\epsilon}_{\theta}(z_{\sigma},\sigma)-\bm{\epsilon}\right\|^{2}
Rectified flowrs=(1−s) r0+s ϵr_{s}=(1-s)\,r_{0}+s\,\bm{\epsilon}vθ(rs,s)\mathbf{v}_{\theta}(r_{s},s)−vθ(rs,s)(1−s)−rss\frac{-\mathbf{v}_{\theta}(r_{s},s)(1-s)-r_{s}}{s}−ϵs-\frac{\bm{\epsilon}}{s}1−ssEr0∼p0Ers∼ps(⋅∣r0)∥ϵ−r0−vθ(rs,s)∥2\frac{1-s}{s}\mathbb{E}_{\mathbf{r}_{0}\sim p_{0}}\mathbb{E}_{\mathbf{r}_{s}\sim p_{s}(\cdot\mid\mathbf{r}_{0})}\left\|\bm{\epsilon}-r_{0}-\mathbf{v}_{\theta}(r_{s},s)\right\|^{2}
Table 6: Training targets and losses under different parameterizations.
Model TypeForward ProcessReverse ProcessLoss (up to a weight factor)
VP-SDExt=αt x0+1−αt ϵx_{t}=\sqrt{\alpha_{t}}\,x_{0}+\sqrt{1-\alpha_{t}}\,\bm{\epsilon}dxt′=[12xt′+s(xt′,T−t′)]dt′+dWt′dx_{t^{\prime}}=\left[\frac{1}{2}x_{t^{\prime}}+\mathbf{s}(x_{t^{\prime}},T-t^{\prime})\right]dt^{\prime}+dW_{t^{\prime}}Ex0∼p0Ext∼pt(⋅∣x0)∥−ϵ1−αt−sθ(xt,t)∥2\mathbb{E}_{\mathbf{x}_{0}\sim p_{0}}\mathbb{E}_{\mathbf{x}_{t}\sim p_{t}(\cdot\mid\mathbf{x}_{0})}\left\|-\frac{\bm{\epsilon}}{\sqrt{1-\alpha_{t}}}-\mathbf{s}_{\theta}(x_{t},t)\right\|^{2}
VP-ODExt=αt x0+1−αt ϵx_{t}=\sqrt{\alpha_{t}}\,x_{0}+\sqrt{1-\alpha_{t}}\,\bm{\epsilon}dxt′=12[xt′+s(xt′,T−t′)]dt′dx_{t^{\prime}}=\frac{1}{2}\left[x_{t^{\prime}}+\mathbf{s}(x_{t^{\prime}},T-t^{\prime})\right]dt^{\prime}Ex0∼p0Ext∼pt(⋅∣x0)∥−ϵ1−αt−sθ(xt,t)∥2\mathbb{E}_{\mathbf{x}_{0}\sim p_{0}}\mathbb{E}_{\mathbf{x}_{t}\sim p_{t}(\cdot\mid\mathbf{x}_{0})}\left\|-\frac{\bm{\epsilon}}{\sqrt{1-\alpha_{t}}}-\mathbf{s}_{\theta}(x_{t},t)\right\|^{2}
VE-Karraszσ=z0+σ ϵz_{\sigma}=z_{0}+\sigma\,\bm{\epsilon}dzσ′=−ϵ(zσ′,Σ−σ′) dσ′dz_{\sigma^{\prime}}=-\bm{\epsilon}(z_{\sigma^{\prime}},\Sigma-\sigma^{\prime})\,d\sigma^{\prime}Ez0∼p0Ezσ∼pσ(⋅∣z0)∥ϵθ(zσ,σ)−ϵ∥2\mathbb{E}_{\mathbf{z}_{0}\sim p_{0}}\mathbb{E}_{\mathbf{z}_{\sigma}\sim p_{\sigma}(\cdot\mid\mathbf{z}_{0})}\left\|\bm{\epsilon}_{\theta}(z_{\sigma},\sigma)-\bm{\epsilon}\right\|^{2}
Rectified flowrs=(1−s) r0+s ϵr_{s}=(1-s)\,r_{0}+s\,\bm{\epsilon}drs′=−v(rs′,1−s′) ds′dr_{s^{\prime}}=-\mathbf{v}(r_{s^{\prime}},1-s^{\prime})\,ds^{\prime}Er0∼p0Ers∼ps(⋅∣r0)∥ϵ−r0−vθ(rs,s)∥2\mathbb{E}_{\mathbf{r}_{0}\sim p_{0}}\mathbb{E}_{\mathbf{r}_{s}\sim p_{s}(\cdot\mid\mathbf{r}_{0})}\left\|\bm{\epsilon}-r_{0}-\mathbf{v}_{\theta}(r_{s},s)\right\|^{2}
Table 7: Unified summary of forward process, reverse process, and objective.

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