Denoising Diffusion Probabilistic Models

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Diffusion models at a glance

A denoising diffusion model is built from two coupled Markov chains running in opposite directions over the same set of intermediate states $x_0, x_1, \ldots, x_T$:

• Forward diffusion process (fixed). Gradually corrupts a clean data point $x_0$ into pure Gaussian noise $x_T$ by adding a small amount of Gaussian noise at each step. No learned parameters; specified entirely by a noise schedule $\beta_t$. Read it as a stack of fixed VAE encoders.
• Reverse denoising process (learnable). Starts from pure noise $x_T \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$ and gradually denoises down to $x_0$. Read it as a stack of learnable VAE decoders — but with a single neural network shared across all $T$ timesteps and conditioned on $t$.

Pictorially, both chains run over the same intermediate states: Forward q (fixed): gradually add noise Editable Mermaid source: images/forward-chain.mermaid.md Filmstrip from CMU 11-785, Lecture 24 (Diffusion), slide 24. Reverse pθ (learned): gradually denoise Editable Mermaid source: images/reverse-chain.mermaid.md Filmstrip from CMU 11-785, Lecture 24 (Diffusion), slide 27. Generation = run the reverse chain. Training = teach the reverse chain to undo a step of the forward chain. The rest of this page builds both chains from a single mathematical primitive: Bishop’s linear-Gaussian theorem.

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Bishop’s linear-Gaussian theorem applied to one diffusion step

A single noising step couples $x_{t-1}$ and $x_t$ through a linear-Gaussian relationship, an instance of Bishop’s linear-Gaussian theorem on the gaussians prerequisite page. Recognizing the kernel as a special case of Bishop’s template lets us read off the marginal in closed form and gives a clean route to the multi-step closed-form jump $q(x_t \mid x_0)$ in the next section. Specialize the theorem to one DDPM step. Take $x = x_{t-1}$, $y = x_t$, with prior $x_{t-1} \sim \mathcal{N}(\mu_{t-1}, \Sigma_{t-1})$. The forward kernel is the linear-Gaussian conditional $$q(x_t \mid x_{t-1}) = \mathcal{N}\left(x_t \mid \sqrt{1-\beta_t}\, x_{t-1},\; \beta_t I\right),$$ which matches Bishop’s $\mathbf{y} = \mathbf{A}\mathbf{x} + \mathbf{b} + \boldsymbol{\varepsilon}$ template with $\mathbf{A} = \sqrt{1-\beta_t}\, I$, $\mathbf{b} = 0$, conditional precision $\mathbf{L} = \beta_t^{-1} I$, and prior precision $\boldsymbol{\Lambda} = \Sigma_{t-1}^{-1}$. Plugging into Bishop’s marginal formula gives the forward marginal $$q(x_t) = \mathcal{N}\left(x_t \mid \sqrt{1-\beta_t}\, \mu_{t-1},\; \beta_t I + (1-\beta_t)\, \Sigma_{t-1}\right).$$ This is the workhorse. Every later result composes the forward kernel: the full forward chain $q(x_{1:T} \mid x_0)$, the closed-form one-shot jump $q(x_t \mid x_0)$, and the forward posterior $q(x_{t-1} \mid x_t, x_0)$ that turns out to be the right reverse-direction target — exact in closed form for any data distribution, and the object the learned $p_\theta$ will be trained to imitate. The next two sections build out the forward chain; after that the learned $p_\theta$ is introduced.

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The forward process: composing $T$ steps

The single-step kernel $q(x_t \mid x_{t-1})$, repeated $t$ times with a fixed schedule $\beta_1, \ldots, \beta_T \in (0, 1)$, defines the full forward chain: $$q(x_{1:T} \mid x_0) \;=\; \prod_{t=1}^{T} q(x_t \mid x_{t-1}).$$ Closed-form jump from $x_0$ to any $x_t$. Because each step is linear-Gaussian and they compose, you can sample $x_t$ in a single shot without simulating the intermediate states. Define $\alpha_t = 1 - \beta_t$ and the cumulative product $\bar\alpha_t = \prod_{s=1}^t \alpha_s$. Iterating Bishop’s per-step recursion across $t$ steps gives $$\begin{aligned} q(x_t \mid x_0) &= \mathcal{N}\!\left(x_t;\; \sqrt{\bar\alpha_t}\, x_0,\; (1 - \bar\alpha_t)\, \mathbf{I}\right), \\ x_t &= \sqrt{\bar\alpha_t}\, x_0 \;+\; \sqrt{1 - \bar\alpha_t}\,\epsilon, \quad \epsilon \sim \mathcal{N}(\mathbf{0}, \mathbf{I}). \end{aligned}$$ This identity is what makes DDPM training cheap: at every gradient step you draw a random $t$ and compute $x_t$ directly from $x_0$ rather than rolling out the chain. The schedule is designed so that $\bar\alpha_T \to 0$, which makes $q(x_T \mid x_0) \approx \mathcal{N}(\mathbf{0}, \mathbf{I})$ regardless of $x_0$. The endpoint of the forward chain is therefore (approximately) data-independent pure noise — the same prior the reverse chain will start from.

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The reverse process: parameterizing what $q$ hides

To generate, we want to draw $x_T \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$ and then iteratively sample $$x_{t-1} \;\sim\; q(x_{t-1} \mid x_t), \qquad t = T, T-1, \ldots, 1.$$ The catch: $q(x_{t-1} \mid x_t)$ as a function of $x_t$ alone is not directly tractable at sampling time. Even though $x_{t-1}$ and $x_t$ are jointly distributed under the forward joint, computing this conditional requires marginalizing over the data distribution $q(x_0)$ — which is precisely what we are trying to model. DDPM’s modeling choice: approximate the reverse step with a learned Gaussian, $$p_\theta(x_{t-1} \mid x_t) \;=\; \mathcal{N}\!\left(x_{t-1};\; \mu_\theta(x_t, t),\; \sigma_t^2\, \mathbf{I}\right).$$ Two reasons this functional form is reasonable:

1. Small-$\beta_t$ limit. When each forward step injects only a little noise, the true reverse conditional $q(x_{t-1} \mid x_t)$ — even with a non-Gaussian data distribution — is well-approximated by a Gaussian (Sohl-Dickstein et al. 2015). That is the local justification for choosing a Gaussian functional form for $p_\theta$.
2. Conditioned on $x_0$, $q$ is genuinely Gaussian. A close cousin, $q(x_{t-1} \mid x_t, x_0)$, is exactly Gaussian for any data distribution $q(x_0)$ — Bayes’ rule on the forward joint with $x_0$ as a fixed parameter (full derivation in the Notes on the reverse conditional deep-dive below). Training will use this two-conditional form (which has $x_0$ available) as the target the learned $p_\theta$ should match.

Two notes on the parameters:

• $\mu_\theta(x_t, t)$ is the only learned quantity. A single neural network produces it for every $t$, with $t$ fed in through a sinusoidal time embedding so the same weights handle every noise level.
• $\sigma_t^2$ is typically fixed by the schedule, commonly either $\beta_t$ or the DDPM posterior variance $\tilde\beta_t$ from $q(x_{t-1} \mid x_t, x_0)$, so the variance is not learned in the basic DDPM.

The full reverse joint factorizes as $$p_\theta(x_{0:T}) \;=\; p(x_T)\, \prod_{t=1}^{T} p_\theta(x_{t-1} \mid x_t), \qquad p(x_T) = \mathcal{N}(\mathbf{0}, \mathbf{I}),$$ and sampling is the obvious thing: draw $x_T$ from the standard Gaussian, then walk one step at a time toward $x_0$.

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Notes on the reverse conditional

With both chains in place, three observations are worth flagging — one notational, two structural. The deep-dive at the end is safe to skip on a first reading.

1. Notation: what $q$ does and does not mean. The symbol $q$ does not tag a direction in time. The forward process defines a joint distribution $q(x_0, x_1, \ldots, x_T) = q(x_0) \prod_{t=1}^{T} q(x_t \mid x_{t-1})$, and anything you can compute from that joint stays inside the q-family. Both $q(x_t \mid x_{t-1})$ and its time-reversal $q(x_{t-1} \mid x_t)$ live in it. But only some of these conditionals stay analytically simple after marginalizing over the data distribution: $q(x_t \mid x_{t-1})$ is fixed and explicit by construction, while $q(x_{t-1} \mid x_t)$ is generally not available in closed form unless extra Gaussian assumptions are imposed. The contrast is with $p_\theta(x_{t-1} \mid x_t)$, which is the learned generative model used at sampling time, where $x_0$ (and therefore the data distribution) is no longer available.
2. What $p_\theta$ targets. The training target the learned $p_\theta(x_{t-1} \mid x_t)$ is fitted against is the forward posterior $q(x_{t-1} \mid x_t, x_0)$, which is exactly Gaussian for any data distribution (see the deep-dive below). At sampling time $x_0$ is unavailable, so the network’s job is to predict what that posterior would have said using only $x_t$ and $t$.
3. Variance accumulates predictably. Each forward step adds $\beta_t I$ to a $(1-\beta_t)$-shrunken copy of $\Sigma_{t-1}$. The closed-form jump $q(x_t \mid x_0) = \mathcal{N}(\sqrt{\bar\alpha_t}\, x_0,\, (1 - \bar\alpha_t)\, I)$ in the forward-process section is precisely the iteration of this rule across $T$ steps.

Optional deep-dive: why q(xₜ₋₁ ∣ xₜ, x₀) is Gaussian for any data distribution

The naive reverse conditional $q(x_{t-1} \mid x_t)$ — conditioning only on the current noisy state — is not in closed form for an arbitrary data distribution: it requires marginalizing over $q(x_0)$, which is exactly what we are trying to model. The standard DDPM derivation (Ho et al. 2020, eq. 6-7) sidesteps that by working with $q(x_{t-1} \mid x_t, x_0)$ — conditioning on both the current noisy state and the original clean sample — which is exactly Gaussian for any $q(x_0)$.The two-conditional form is Gaussian for any data distribution $q(x_0)$. Apply Bayes’ rule on the forward joint:$$q(x_{t-1} \mid x_t, x_0) \;=\; \frac{q(x_t \mid x_{t-1}, x_0)\, q(x_{t-1} \mid x_0)}{q(x_t \mid x_0)}.$$By the Markov property of the forward chain, $q(x_t \mid x_{t-1}, x_0) = q(x_t \mid x_{t-1})$ is the Gaussian noising kernel. The other two terms are the closed-form jumps from $x_0$ derived in the forward-process section, both Gaussian. The product/quotient of three Gaussians (in $x_{t-1}$) is again Gaussian. The data distribution $q(x_0)$ never enters because $x_0$ is a fixed parameter here, not a random variable being marginalized over — its value just shifts the means.By contrast, $q(x_{t-1} \mid x_t)$ without $x_0$ marginalizes $x_0$ out:$$q(x_{t-1} \mid x_t) \;=\; \int q(x_{t-1} \mid x_t, x_0)\, q(x_0 \mid x_t)\, dx_0,$$a mixture of Gaussians weighted by the posterior $q(x_0 \mid x_t)$, generally non-Gaussian unless $q(x_0)$ itself is Gaussian. The two-conditional form $q(x_{t-1} \mid x_t, x_0)$ avoids this entirely: with $x_0$ as a fixed parameter, the data distribution drops out of the algebra and the closed form holds regardless of $q(x_0)$. That is the forward posterior that appears in the ELBO derivation below — at training time you have $x_0$ available, and the result is a Gaussian whose mean and variance are explicit functions of $(x_t, x_0, t)$ alone.

Optional worked example: a three-step diffusion run end-to-end

Consider a toy diffusion model with only three noising steps:$$x_0 \;\to\; x_1 \;\to\; x_2 \;\to\; x_3.$$$x_0$ is a clean data sample (for example, an image) and $x_3$ is almost Gaussian noise. Pick a forward noise schedule$$\beta_1, \beta_2, \beta_3 \in (0, 1),$$and define $\alpha_t = 1 - \beta_t$ together with the cumulative product$$\bar\alpha_t = \prod_{s=1}^{t} \alpha_s, \qquad\text{so}\qquad \bar\alpha_1 = \alpha_1,\quad \bar\alpha_2 = \alpha_1 \alpha_2,\quad \bar\alpha_3 = \alpha_1 \alpha_2 \alpha_3.$$Forward process. The forward process is fixed: it gradually corrupts the data by$$q(x_t \mid x_{t-1}) = \mathcal{N}\left(x_t;\; \sqrt{\alpha_t}\, x_{t-1},\; \beta_t I\right).$$For the three steps,$$q(x_1 \mid x_0) = \mathcal{N}\left(x_1;\; \sqrt{\alpha_1}\, x_0,\; \beta_1 I\right),$$$$q(x_2 \mid x_1) = \mathcal{N}\left(x_2;\; \sqrt{\alpha_2}\, x_1,\; \beta_2 I\right),$$$$q(x_3 \mid x_2) = \mathcal{N}\left(x_3;\; \sqrt{\alpha_3}\, x_2,\; \beta_3 I\right).$$Equivalently, you can sample any noisy point directly from $x_0$:$$q(x_t \mid x_0) = \mathcal{N}\left(x_t;\; \sqrt{\bar\alpha_t}\, x_0,\; (1 - \bar\alpha_t)\, I\right),$$so the final noisy sample is$$x_3 = \sqrt{\bar\alpha_3}\, x_0 + \sqrt{1 - \bar\alpha_3}\, \epsilon, \qquad \epsilon \sim \mathcal{N}(0, I).$$Reverse process. The reverse process tries to undo the corruption: $x_3 \to x_2 \to x_1 \to x_0$. Parameterize the learned reverse kernel as$$p_\theta(x_{t-1} \mid x_t) = \mathcal{N}\left(x_{t-1};\; \mu_\theta(x_t),\; \Sigma_\theta(x_t)\right),$$instantiated at the three steps as $p_\theta(x_2 \mid x_3)$, $p_\theta(x_1 \mid x_2)$, $p_\theta(x_0 \mid x_1)$.A note on notation: $\mu_\theta(x_t)$ and $\epsilon_\theta(x_t)$ are written with a single argument because $t$ is already pinned down by the subscript on $x_t$. In code, $\mu_\theta$ and $\epsilon_\theta$ are a single shared network used at every timestep, conditioned on $t$ through a learned time embedding; the $t$ argument is implicit in the input typing.In DDPM the network predicts the noise $\epsilon_\theta(x_t)$ that was added. Plugging that into the reverse mean gives$$\mu_\theta(x_t) = \frac{1}{\sqrt{\alpha_t}}\left( x_t - \frac{\beta_t}{\sqrt{1 - \bar\alpha_t}}\, \epsilon_\theta(x_t) \right).$$Starting from $x_3 \sim \mathcal{N}(0, I)$, the three reverse transitions are$$\mu_\theta(x_3) = \frac{1}{\sqrt{\alpha_3}}\left( x_3 - \frac{\beta_3}{\sqrt{1 - \bar\alpha_3}}\, \epsilon_\theta(x_3) \right), \qquad x_2 \sim \mathcal{N}\left(\mu_\theta(x_3),\; \Sigma_\theta(x_3)\right),$$$$\mu_\theta(x_2) = \frac{1}{\sqrt{\alpha_2}}\left( x_2 - \frac{\beta_2}{\sqrt{1 - \bar\alpha_2}}\, \epsilon_\theta(x_2) \right), \qquad x_1 \sim \mathcal{N}\left(\mu_\theta(x_2),\; \Sigma_\theta(x_2)\right),$$$$\mu_\theta(x_1) = \frac{1}{\sqrt{\alpha_1}}\left( x_1 - \frac{\beta_1}{\sqrt{1 - \bar\alpha_1}}\, \epsilon_\theta(x_1) \right), \qquad x_0 \sim \mathcal{N}\left(\mu_\theta(x_1),\; \Sigma_\theta(x_1)\right).$$The essential idea: the forward chain adds known Gaussian noise, and the reverse chain learns how much noise to remove. Putting both directions on the same picture,$$x_0 \xrightarrow{q} x_1 \xrightarrow{q} x_2 \xrightarrow{q} x_3, \qquad x_3 \sim \mathcal{N}(0, I) \xrightarrow{p_\theta} x_2 \xrightarrow{p_\theta} x_1 \xrightarrow{p_\theta} x_0.$$

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What the network outputs

The reverse-step distribution is parameterized as $$p_\theta(x_{t-1} \mid x_t) = \mathcal{N}\!\left(x_{t-1};\, \mu_\theta(x_t, t),\, \sigma_t^2\, \mathbf{I}\right),$$ but what the neural network literally computes is a design choice. There are three algebraically equivalent options:

1. Predict the mean directly — $\mu_\theta(x_t, t)$. The most direct read of the parameterization above; the network output is a vector with the shape of $x_t$, used as the mean of the reverse Gaussian.
2. Predict the noise — $\epsilon_\theta(x_t, t)$. The network outputs a vector with the shape of $x_t$ that estimates the noise that was added when forming $x_t$ from $x_0$ via the closed-form jump. The reverse-step mean is then derived analytically: $$\mu_\theta(x_t, t) = \frac{1}{\sqrt{\alpha_t}}\!\left( x_t - \frac{\beta_t}{\sqrt{1 - \bar\alpha_t}}\, \epsilon_\theta(x_t, t) \right).$$
3. Predict the clean data — $\hat{x}_{0,\theta}(x_t, t)$. The network outputs an estimate of the original $x_0$. The reverse-step mean is derived from the forward posterior $q(x_{t-1} \mid x_t, x_0)$ with the network’s $\hat{x}_0$ plugged in for the unknown $x_0$.

These three are interchangeable under the reparameterization $x_t = \sqrt{\bar\alpha_t}\, x_0 + \sqrt{1 - \bar\alpha_t}\, \epsilon$:

• Given $(x_t, \hat\epsilon)$: solve for $\hat{x}_0 = (x_t - \sqrt{1 - \bar\alpha_t}\, \hat\epsilon) \,/\, \sqrt{\bar\alpha_t}$.
• Given $(x_t, \hat{x}_0)$: solve for $\hat\epsilon = (x_t - \sqrt{\bar\alpha_t}\, \hat{x}_0) \,/\, \sqrt{1 - \bar\alpha_t}$.

DDPM picks ε-prediction. Three reasons make it empirically dominant:

• The training target $\epsilon \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$ has fixed scale at every $t$, so the regression problem is well-conditioned across all noise levels.
• The corresponding training loss collapses to a plain unweighted MSE — the $L_\text{simple}$ derived in the next section.
• The network’s job becomes a single intuitive task: “look at this noisy data and tell me the noise that was added”.

Concretely, one forward pass of the DDPM network looks like: Editable Mermaid source: images/network-io.mermaid.md The data input dimension is whatever $x_t$ has (2 for the MoG example, 3×H×W for images). The timestep enters through a sinusoidal time embedding (the original DDPM lifts Vaswani’s positional-encoding formula and feeds the integer $t$ into it) so a single set of weights handles every noise level. At sampling time, the predicted noise gets plugged into the analytic reverse-step mean above, and a small amount of fresh Gaussian noise is added (controlled by $\sigma_t^2$) to draw $x_{t-1}$. The next section derives why training this network on the noise-prediction MSE is exactly the right loss to maximize the data likelihood.

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Training objective: from VAE ELBO to the DDPM loss

You now have a fixed forward chain $q$ and a parameterized reverse chain $p_\theta$. The remaining question is what loss to train $\theta$ on. The answer is the same Evidence Lower Bound (ELBO) derived for VAEs (see Optimization and the ELBO for the single-latent derivation), applied here to a deep, Markov latent chain whose encoder happens to be fixed. DDPM is a hierarchical VAE with two simplifying choices: the latent is a Markov chain $\mathbf x_{1:T}$, and the encoder is fixed, namely hand-designed Gaussian noise injection with no learnable parameters. Hierarchical here means a stack of $T$ latents $\mathbf x_1, \ldots, \mathbf x_T$ rather than the single latent $\mathbf z$ from the basic VAE architecture. A generic hierarchical VAE (NVAE, ladder-VAE, ResNet-VAE) learns both the top-down decoder $p_\theta(\mathbf x_{t-1} \mid \mathbf x_t)$ and a bottom-up encoder $q_\phi(\mathbf x_t \mid \mathbf x_{t-1}, \mathbf x_0)$ at every level; DDPM keeps the top-down chain learnable and freezes the bottom-up chain to a fixed Gaussian schedule. Substitute these into the joint-form VAE ELBO and you get exactly DDPM’s eq. (3): $\log p_\theta(\mathbf x_0) \;\ge\; \mathbb{E}_{q(\mathbf x_{1:T} | \mathbf x_0)}\left[\log \frac{p_\theta(\mathbf x_{0:T})}{q(\mathbf x_{1:T} | \mathbf x_0)}\right] \;=:\; -L.$ The DDPM training loss $L$ is the negative VAE ELBO with the single latent replaced by the entire noise chain.

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A telescoping decomposition

Both chains are Markov, so the joint distributions factorize across $t$. Applying Bayes’ rule to rewrite each forward step in terms of the forward posterior $q(\mathbf x_{t-1} | \mathbf x_t, \mathbf x_0)$ telescopes the bound into three pieces (Ho et al., eq. 5): $L \;=\; \mathbb{E}_q\left[\,\underbrace{KL\left(q(\mathbf x_T | \mathbf x_0) \,\|\, p(\mathbf x_T)\right)}_{L_T} \;+\; \sum_{t > 1} \underbrace{KL\left(q(\mathbf x_{t-1} | \mathbf x_t, \mathbf x_0) \,\|\, p_\theta(\mathbf x_{t-1} | \mathbf x_t)\right)}_{L_{t-1}} \;-\; \underbrace{\log p_\theta(\mathbf x_0 | \mathbf x_1)}_{L_0}\,\right]$ Each piece has a direct VAE counterpart:

• $L_0 = -\log p_\theta(\mathbf x_0 | \mathbf x_1)$ is the reconstruction term, playing the same role as $-\log p_\theta(\mathbf x | \mathbf z)$ in the single-latent VAE, just at the bottom rung of the chain instead of after one decoder pass.
• $L_T = KL\left(q(\mathbf x_T | \mathbf x_0) \,\|\, p(\mathbf x_T)\right)$ is the prior-matching term, the same role as $KL(q_\phi(\mathbf z | \mathbf x) \,\|\, p(\mathbf z))$ in the VAE. Because the forward process is fixed and the schedule $\beta_t$ is chosen so that $q(\mathbf x_T | \mathbf x_0) \to \mathcal N(\mathbf 0, \mathbf I)$ for large $T$, this term has no parameters to optimize: it is approximately constant during training, and DDPM drops it from the loss.
• $L_{t-1}$ are the per-step transition KLs between the analytically tractable forward posterior $q(\mathbf x_{t-1} | \mathbf x_t, \mathbf x_0)$, whose closed form was derived in Bishop’s linear-Gaussian theorem applied to one diffusion step near the top of this page, and the learned reverse step $p_\theta(\mathbf x_{t-1} | \mathbf x_t)$. These have no counterpart in a single-latent VAE; they appear because the chain has $T$ rungs.

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Why this is operationally simpler than a hierarchical VAE

Fixing the encoder buys two simplifications that ordinary VAEs cannot exploit:

1. No encoder gradient. The forward process has no $\phi$. The bound-tightening role the encoder plays in a VAE (gradients on $\phi$ minimizing the posterior KL) disappears entirely. Training is a single-network problem in $\theta$.
2. All $L_{t-1}$ terms share parameters. In the standard fixed-variance DDPM setup, both $q(\mathbf x_{t-1} | \mathbf x_t, \mathbf x_0)$ and $p_\theta(\mathbf x_{t-1} | \mathbf x_t)$ are Gaussians with matched prescribed covariance, so the learnable part is the mean. Combining the reparameterization $\mathbf x_t = \sqrt{\bar\alpha_t}\, \mathbf x_0 + \sqrt{1-\bar\alpha_t}\, \boldsymbol\epsilon$ from the worked example above with the noise-prediction parameterization $\mu_\theta(\mathbf x_t) \;=\; \frac{1}{\sqrt{\alpha_t}}\left(\mathbf x_t - \frac{\beta_t}{\sqrt{1 - \bar\alpha_t}}\, \boldsymbol\epsilon_\theta(\mathbf x_t)\right)$ collapses each $L_{t-1}$, up to a $t$-dependent weight, to a noise-prediction MSE (Ho et al., §3.2): $L_{t-1} \;\propto\; \mathbb{E}_{\mathbf x_0, \boldsymbol\epsilon}\left[\,\bigl\|\boldsymbol\epsilon \,-\, \boldsymbol\epsilon_\theta\bigl(\sqrt{\bar\alpha_t}\, \mathbf x_0 + \sqrt{1-\bar\alpha_t}\, \boldsymbol\epsilon,\, t\bigr)\bigr\|^2\,\right].$ Dropping the $t$-dependent weight gives the simple training objective $L_{\text{simple}}$ that produced DDPM’s image-quality results.

When you train DDPM on the noise-prediction MSE in the DDPM MoG example, you are optimizing a re-weighted version of the same VAE ELBO, applied to a $T$-deep latent chain whose encoder you decided in advance instead of learning. The score-based methods page reaches the same training loss from a different door (denoising score matching), and the equivalence between the noise predictor $\boldsymbol\epsilon_\theta$ and the score $\nabla_{\mathbf x} \log p_t(\mathbf x)$ is what unifies both views.

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References

1. Kingma, Welling. Auto-Encoding Variational Bayes. ICLR 2014. arxiv.org/abs/1312.6114
2. Ho, Jain, Abbeel. Denoising Diffusion Probabilistic Models. NeurIPS 2020. arxiv.org/abs/2006.11239
3. Sohl-Dickstein et al. Deep Unsupervised Learning using Nonequilibrium Thermodynamics. ICML 2015. arxiv.org/abs/1503.03585
4. Song et al. Score-Based Generative Modeling through Stochastic Differential Equations. ICLR 2021. arxiv.org/abs/2011.13456

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