Direct Denoising Diffusion

Recover Capacity by Direct Denoising—Predict Clean Data Not Noise

Diffusion models typically predict noise (epsilon) or velocity at each denoising step. This choice seems natural but imposes a hidden cost: noise and velocity distributions span the full data dimensionality, wasting model capacity on high-dimensional structure. Direct x-prediction assumes natural data lies on a low-dimensional manifold—the model only needs to preserve essential information while filtering noise, using capacity more efficiently.

This paper demonstrates that limited-capacity networks can generate high-dimensional data via x-prediction where epsilon/v-prediction fails. The insight is practical: shift the prediction target to align with your data's intrinsic dimensionality, not the ambient space.

Core Concept

Diffusion models work by iteratively adding and removing noise. At each step, the model must predict either:

1. Noise (ε-prediction): The random noise added; distributed across full dimensionality
2. Velocity (v-prediction): A mixture; also spreads across full space
3. Clean data (x-prediction): The denoised output; concentrated on low-dimensional manifold

The mathematical relationships between these are equivalent—they're coordinate transformations of the same underlying process. However, they differ fundamentally in capacity requirement:

• High-dimensional noise: Requires network to preserve all directional information; capacity-intensive
• Low-dimensional data: Network only needs to capture essential structure; more efficient

This manifold assumption—natural data lives in lower-dimensional space than noise—creates practical capacity gaps observable at scale.

Architecture Overview

• Manifold Assumption: Natural data concentrates on low-dimensional manifold; noise fills full high-dimensional space
• x-Prediction Head: Network outputs clean data directly, not noise or velocity
• Matching Framework: Use standard diffusion loss (MSE) adapted for x-prediction via mathematical conversion
• Patch-Based Encoding: For image/video models, use large patches to reduce dimensionality and align with manifold structure
• Capacity Scaling: Lower capacity networks exhibit larger performance gaps; x-prediction shows more benefit at limited budgets

Implementation Steps

Step 1: Reformulate Diffusion Process for x-Prediction. Convert noise schedule to x-prediction formulation.

import torch
import torch.nn as nn

class XPredictionDiffusion:
    """
    Diffusion process using x-prediction (direct clean data prediction).
    """
    def __init__(self, timesteps=1000, beta_start=0.0001, beta_end=0.02):
        self.timesteps = timesteps

        # Standard beta schedule
        betas = torch.linspace(beta_start, beta_end, timesteps)
        alphas = 1.0 - betas
        alphas_cumprod = torch.cumprod(alphas, dim=0)

        # Key quantities for x-prediction
        # At step t, noisy image: x_t = sqrt(alpha_bar_t) * x_0 + sqrt(1 - alpha_bar_t) * epsilon
        self.register_buffer('alphas_cumprod', alphas_cumprod)
        self.register_buffer('sqrt_alphas_cumprod', torch.sqrt(alphas_cumprod))
        self.register_buffer('sqrt_1m_alphas_cumprod', torch.sqrt(1.0 - alphas_cumprod))

    def add_noise(self, x_0, t, noise=None):
        """
        Add noise to clean sample (forward process).
        x_0: clean data (batch, channels, height, width)
        t: timestep (batch,)
        """
        if noise is None:
            noise = torch.randn_like(x_0)

        sqrt_alphas = self.sqrt_alphas_cumprod[t]
        sqrt_1m_alphas = self.sqrt_1m_alphas_cumprod[t]

        # Reshape for broadcasting
        while len(sqrt_alphas.shape) < len(x_0.shape):
            sqrt_alphas = sqrt_alphas.unsqueeze(-1)
            sqrt_1m_alphas = sqrt_1m_alphas.unsqueeze(-1)

        # Noisy sample
        x_t = sqrt_alphas * x_0 + sqrt_1m_alphas * noise

        return x_t, noise

    def predict_x0_from_xt(self, x_t, t, model_prediction):
        """
        In x-prediction, the model directly outputs x_0 estimate.
        This is the reverse process; often called "x-denoising".
        """
        # The model prediction IS the x_0 estimate
        return model_prediction

    def loss_x_prediction(self, x_0, t, model_pred):
        """
        L2 loss for x-prediction: MSE between model output and true x_0.
        Simpler than noise prediction (no noise extraction needed).
        """
        return torch.nn.functional.mse_loss(model_pred, x_0, reduction='mean')

Step 2: Build Network with x-Prediction Head. Design model to output clean data directly.

class XPredictionModel(nn.Module):
    """
    Diffusion model with x-prediction head.
    Can be Transformer, UNet, etc.; example: Transformer-based.
    """
    def __init__(self, input_channels=3, patch_size=4, hidden_dim=768, num_layers=12):
        super().__init__()
        self.patch_size = patch_size
        self.hidden_dim = hidden_dim

        # Patch embedding
        self.patch_embed = nn.Linear(3 * patch_size * patch_size, hidden_dim)

        # Time embedding
        self.time_embed = nn.Sequential(
            nn.Linear(1, hidden_dim // 2),
            nn.GELU(),
            nn.Linear(hidden_dim // 2, hidden_dim)
        )

        # Transformer backbone
        self.transformer = nn.TransformerEncoder(
            nn.TransformerEncoderLayer(hidden_dim, nhead=8, dim_feedforward=3072),
            num_layers=num_layers
        )

        # x-Prediction head: output clean data patches
        self.x_pred_head = nn.Linear(hidden_dim, 3 * patch_size * patch_size)

    def forward(self, x_t, t):
        """
        x_t: noisy input (batch, 3, H, W)
        t: timestep (batch,)
        Returns: predicted clean image (batch, 3, H, W)
        """
        batch, channels, height, width = x_t.shape

        # Patchify
        patches = self._patchify(x_t)  # (batch, num_patches, patch_dim)

        # Embed patches
        patch_embeds = self.patch_embed(patches)  # (batch, num_patches, hidden_dim)

        # Time embedding
        time_emb = self.time_embed(t.unsqueeze(-1).float())  # (batch, hidden_dim)
        time_emb = time_emb.unsqueeze(1)  # (batch, 1, hidden_dim)

        # Add time to patches
        patch_embeds = patch_embeds + time_emb  # Broadcasting

        # Transformer
        hidden = self.transformer(patch_embeds)  # (batch, num_patches, hidden_dim)

        # x-Prediction: directly output clean patches
        clean_patches = self.x_pred_head(hidden)  # (batch, num_patches, patch_dim)

        # Unpatchify
        x_0_pred = self._unpatchify(clean_patches, height, width)

        return x_0_pred

    def _patchify(self, x):
        """Convert image to non-overlapping patches."""
        batch, channels, height, width = x.shape
        x = x.reshape(
            batch,
            channels,
            height // self.patch_size,
            self.patch_size,
            width // self.patch_size,
            self.patch_size
        )
        x = x.permute(0, 2, 4, 1, 3, 5).contiguous()
        x = x.reshape(batch, -1, channels * self.patch_size * self.patch_size)
        return x

    def _unpatchify(self, patches, height, width):
        """Reconstruct image from patches."""
        batch, num_patches, patch_dim = patches.shape
        channels = 3

        patches = patches.reshape(
            batch,
            height // self.patch_size,
            width // self.patch_size,
            channels,
            self.patch_size,
            self.patch_size
        )
        patches = patches.permute(0, 3, 1, 4, 2, 5).contiguous()
        x = patches.reshape(batch, channels, height, width)

        return x

Step 3: Training Loop. Train with x-prediction loss.

def train_x_prediction_diffusion(model, dataloader, num_epochs=100, lr=1e-4):
    """
    Train diffusion model with x-prediction.
    """
    optimizer = torch.optim.Adam(model.parameters(), lr=lr)
    diffusion = XPredictionDiffusion(timesteps=1000)

    for epoch in range(num_epochs):
        total_loss = 0

        for x_0 in dataloader:  # x_0: clean images
            batch_size = x_0.shape[0]

            # Random timesteps
            t = torch.randint(0, diffusion.timesteps, (batch_size,))

            # Forward process: add noise
            x_t, noise = diffusion.add_noise(x_0, t)

            # Model prediction (x-prediction)
            x_0_pred = model(x_t, t)

            # Loss: direct MSE between prediction and clean data
            loss = diffusion.loss_x_prediction(x_0, t, x_0_pred)

            optimizer.zero_grad()
            loss.backward()
            torch.nn.utils.clip_grad_norm_(model.parameters(), 1.0)
            optimizer.step()

            total_loss += loss.item()

        avg_loss = total_loss / len(dataloader)
        print(f"Epoch {epoch}: avg_loss={avg_loss:.4f}")

Step 4: Inference (Denoising). Generate by iteratively denoising.

@torch.no_grad()
def generate_x_prediction(model, diffusion, shape, num_steps=100):
    """
    Generate samples via iterative x-prediction denoising.
    """
    # Start with noise
    x_t = torch.randn(shape)

    # Iterate denoising (reverse process)
    timesteps = torch.linspace(999, 0, num_steps).long()

    for t in timesteps:
        t_batch = torch.full((x_t.shape[0],), t)

        # Model predicts clean image at this step
        x_0_pred = model(x_t, t_batch)

        # Get alphas for current and next step
        alpha_t = diffusion.alphas_cumprod[t]
        alpha_prev = diffusion.alphas_cumprod[t - 1] if t > 0 else 1.0

        # Update x_t using predicted x_0
        # This is the reverse step in x-prediction formulation
        c1 = torch.sqrt((1 - alpha_prev) / (1 - alpha_t))
        c2 = torch.sqrt(alpha_prev / alpha_t)

        x_t = c2 * x_0_pred + c1 * (x_t - torch.sqrt(1 - alpha_t) * x_0_pred)

        # Add minimal noise if not final step
        if t > 0:
            noise = torch.randn_like(x_t)
            sigma_t = torch.sqrt((1 - alpha_prev) / (1 - alpha_t) * (1 - alpha_t / alpha_prev))
            x_t = x_t + sigma_t * noise

    return x_t.clamp(-1, 1)

Practical Guidance

When to Use: Training diffusion models on bounded data (images, quantized latents) with limited model capacity. x-Prediction shows largest benefits when model capacity is restricted (< 1B parameters).

Hyperparameters:

• Patch size: larger patches (8–16) reduce dimensionality and favor x-prediction; adjust based on image resolution
• Schedule: use same beta schedule as noise-pred; mathematical equivalence enables direct comparison
• Learning rate: typically same as noise-pred; no special adjustments needed

Pitfalls:

• Unbounded predictions: Models can output values outside data range; apply clipping or use bounded activations
• Instability at early timesteps: Very noisy inputs may lead to erratic x-0 predictions; use warm-up or gradient clipping
• Not always faster: Wall-clock time may be similar (both models have same architecture); gain is capacity efficiency, not compute
• Dataset-dependent: Manifold assumption stronger on natural images; weaker on synthetic/high-dimensional data

When NOT to Use: Very large models where capacity is not a bottleneck; data not concentrated on low-dimensional manifold.

Integration: Drop-in replacement for noise-prediction models; use same sampling procedures with adapted formulations.

Reference: https://arxiv.org/abs/2511.13720