The denoising inference step in Diffusion-LM is the reverse process of the diffusion model, where the system starts with a sequence of pure Gaussian noise vectors ($x_T$) and iteratively removes the noise step-by-step to recover the continuous word vectors ($x_0$), which are finally converted back into discrete text.

Here is the detailed workflow and mathematics behind this process:

1. The Denoising Transition

At each step $t$ during inference, the model aims to transition from the current noisy state $x_t$ to a slightly less noisy state $x_{t-1}$. This transition is typically modeled as a Gaussian distribution:

$$p_\theta(x_{t-1}|x_t) = \mathcal{N}(x_{t-1}; \mu_\theta(x_t, t), \Sigma_\theta(x_t, t))$$

Instead of predicting the mean $\mu_\theta$ directly, the Diffusion-LM uses a neural network $f_\theta(x_t, t)$ to directly predict the fully denoised state $x_0$ at every step.

2. Computing the Posterior Mean

Once the model has an estimate for $x_0$ via $f_\theta(x_t, t)$, it computes the mean of the tractable Gaussian posterior $q(x_{t-1} | x_0, x_t)$ to sample the next state. The exact closed-form expression for this true posterior mean $\hat{\mu}(x_t, x_0)$ is: $\hat{\mu}(x_t, x_0) = \frac{\sqrt{\bar{\alpha}_{t-1}}\beta_t}{1-\bar{\alpha}_t}x_0 + \frac{\sqrt{\alpha_t}(1-\bar{\alpha}_{t-1})}{1-\bar{\alpha}_t}x_t$.

Definitions of the terms:

• $x_0$: The fully denoised, continuous latent vector (estimated in practice by $f_\theta(x_t, t)$).
• $x_t$: The current noisy continuous vector at step $t$.
• $\beta_t$: The hyperparameter dictating the amount of noise added at diffusion step $t$.
• $\alpha_t$: Defined implicitly as $1 - \beta_t$ (standard in diffusion literature and used in the posterior calculation).
• $\bar{\alpha}_t$: The cumulative product of noise variances up to step $t$, defined mathematically as $\bar{\alpha}_t = \prod_{s=0}^t(1-\beta_s)$.

To perform the actual denoising step, the model substitutes its prediction $f_\theta(x_t, t)$ in place of $x_0$ and samples the new state $x_{t-1}$ from the resulting distribution $q(x_{t-1} | f_\theta(x_t, t), x_t)$.

3. The Clamping Trick

To further reduce rounding errors during inference, Diffusion-LM introduces the “clamping trick”. Instead of using the raw continuous prediction $f_\theta(x_t, t)$, the model maps this vector to its nearest actual discrete word embedding before sampling $x_{t-1}$.

The modified sampling step becomes: $x_{t-1} = \sqrt{\bar{\alpha}} \cdot Clamp(f_\theta(x_t, t)) + \sqrt{1 - \bar{\alpha}}\epsilon$. (Where $\epsilon \sim \mathcal{N}(0, I)$ is standard Gaussian noise).

By clamping the intermediate predictions, the model is forced to commit to specific words earlier in the diffusion process, making the continuous trajectory more precise.

4. Final Rounding to Discrete Text

After completing all denoising steps (typically $T=2000$ or downsampled to $200$ for speed), the model arrives at the final continuous representation $x_0$. The final step is rounding, which maps $x_0$ back to discrete text $w$. This is done by selecting the most probable word for each sequence position using a softmax distribution: $\text{argmax } p_\theta(w|x_0) = \prod_{i=1}^n p_\theta(w_i|x_i)$.