Interpolation and timesteps utility

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Overview

1. Per-timestep loss analysis: Measure noise prediction error across diffusion timesteps
2. DDPM noise interpolation: Generate smooth transitions between random noise vectors
3. DDIM noise interpolation: Deterministic interpolation using DDIM sampling

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Usage

python src/utilities/interpolation_and_timesteps.py

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Prerequisites

• Trained MNIST model at best_model.pt in project root
• MNIST dataset will be downloaded automatically if not present

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Expected outputs

1. interp.png: DDPM stochastic interpolation grid (8×9)
2. interp_ddim.png: DDIM deterministic interpolation grid (8×9)
3. Timestep loss plot: matplotlib figure showing error vs timestep buckets

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Functions

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per_timestep_loss

@torch.no_grad()
def per_timestep_loss(
    diffusion,
    loader,
    num_batches=10,
    buckets=10,
    device='cuda'
) -> torch.Tensor

1. Splits timesteps 0-999 into equal buckets (e.g., 10 buckets = 100 steps each)
2. For each batch:
   • Sample random timesteps t uniformly from [0, T)
   • Add noise: x_t, noise = diffusion.add_noise(x, t)
   • Predict noise: pred = model(x_t, t)
   • Compute per-sample MSE: ((pred - noise)**2).mean(dim=(1,2,3))
   • Accumulate into appropriate bucket based on t
3. Average losses across batches
4. Plot bar chart showing error vs timestep range

losses = per_timestep_loss(
    diffusion,
    train_loader,
    num_batches=20,
    buckets=10,
    device='cuda'
)
# losses[0]: average loss for timesteps 0-99
# losses[1]: average loss for timesteps 100-199
# ...
# losses[9]: average loss for timesteps 900-999

• X-axis: Timestep buckets (e.g., “0-99”, “100-199”, …)
• Y-axis: MSE (ε-prediction error)
• Title: “Noise-prediction error vs timestep”

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sample_from_xt

@torch.no_grad()
def sample_from_xt(diffusion, x_T) -> torch.Tensor

for t in reversed(range(T)):  # T=1000 down to 0
    pred_eps = model(x_t, t)
    # Compute mean of p(x_{t-1} | x_t)
    x_t = (1/√α_t) * (x_t - (β_t / √(1-ᾱ_t)) * pred_eps)
    # Add noise if not final step
    if t > 0:
        x_t += √β_t * ε,  ε ~ N(0, I)

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interpolate_noise_and_generate

@torch.no_grad()
def interpolate_noise_and_generate(
    diffusion,
    n=8,
    steps=7,
    save_path='interp.png'
) -> str

1. Generate two random noise endpoints: z0, z1 ~ N(0, I)
2. Create linear interpolation: z_α = (1-α)z0 + αz1 for α ∈ [0, 1]
3. Sample from each interpolated noise: x = sample_from_xt(diffusion, z_α)
4. Arrange as grid with n rows and steps columns
5. Save to save_path using torchvision.utils.save_image

path = interpolate_noise_and_generate(
    diffusion,
    n=8,
    steps=9,
    save_path='samples/interp.png'
)
# Creates 8x9 grid showing smooth transitions
# Each row is independent interpolation
# Each column represents α from 0.0 to 1.0

Row 1: z0[0] -----(α)-----> z1[0]
Row 2: z0[1] -----(α)-----> z1[1]
...
Row n: z0[n] -----(α)-----> z1[n]
       ↑                      ↑
    α=0.0                  α=1.0

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ddim_sample_from_xt

@torch.no_grad()
def ddim_sample_from_xt(diffusion, x_T) -> torch.Tensor

for t in reversed(range(T)):
    eps_hat = model(x_t, t)
    # Predict x_0 from x_t
    x0_hat = (x_t - √(1-ᾱ_t) * eps_hat) / √ᾱ_t
    # Deterministic update to x_{t-1}
    if t > 0:
        x_t = √ᾱ_{t-1} * x0_hat + √(1-ᾱ_{t-1}) * eps_hat
    else:
        x_t = x0_hat

• Consistency: Same x_T always produces same x_0
• Invertibility: Can encode images back to noise
• Interpolation: Smooth transitions in latent space

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interpolate_noise_and_generate_ddim

@torch.no_grad()
def interpolate_noise_and_generate_ddim(
    diffusion,
    n=8,
    steps=7,
    save_path="interp.png"
) -> str

• Deterministic output: No randomness in reverse process
• Smoother interpolations: Consistent mapping from noise to images
• Better quality: Determinism often produces cleaner transitions

path = interpolate_noise_and_generate_ddim(
    diffusion,
    n=8,
    steps=9,
    save_path='samples/interp_ddim.png'
)
# Creates 8x9 deterministic interpolation grid
# Compare with DDPM version to see smoothness difference

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Main script execution

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Initialization

device = torch.device("cuda" if torch.cuda.is_available() else "cpu")

diffusion = DiffusionProcess(
    image_size=28,
    channels=1,
    hidden_dims=[128, 256, 512],
    device=device,
)

# Load trained model
ckpt_path = "best_model.pt"
diffusion.model.load_state_dict(torch.load(ckpt_path, map_location=device))
diffusion.model.eval()

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Dataset loading

transform = transforms.Compose([
    transforms.ToTensor(),
    transforms.Normalize((0.5,), (0.5,)),
])
dataset = datasets.MNIST(
    root="data",
    train=False,
    download=True,
    transform=transform,
)
loader = DataLoader(dataset, batch_size=64, shuffle=True)

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Analysis and generation

1. Timestep loss diagnostic (optional, displays matplotlib plot):
   _ = per_timestep_loss(
       diffusion, loader, num_batches=10, buckets=10, device=device
   )
   
2. DDPM interpolation grid:
   interpolate_noise_and_generate(
       diffusion,
       n=8,
       steps=9,
       save_path="samples/interp.png",
   )
   
3. DDIM interpolation grid:
   interpolate_noise_and_generate_ddim(
       diffusion,
       n=8,
       steps=9,
       save_path="samples/interp_ddim.png",
   )
   

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Output examples

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Timestep loss plot

      Loss
       ↑
  0.05 |     ╭─╮
       |    ╱   ╲
  0.03 |   ╱     ╲
       |  ╱       ╲___
  0.01 |_╱____________╲_____
       └────────────────────→
       0-99  ...  900-999
            Timestep bucket

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Interpolation grid structure

• Each row: Independent interpolation between two random endpoints
• Left column (α=0.0): First random noise vector endpoint
• Right column (α=1.0): Second random noise vector endpoint
• Middle columns: Linear interpolation in noise space
• Smooth semantic transitions demonstrate learned manifold structure

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DDPM vs DDIM comparison

• Stochastic sampling adds noise at each step
• Slight variations in intermediate steps
• May show more diversity but less smoothness

• Deterministic sampling (η=0)
• Perfectly smooth transitions
• Consistent and reproducible interpolations
• Generally cleaner visual quality

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Use cases

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Diagnostic analysis

• Identify problematic timestep ranges during training
• Validate that model learns all diffusion stages
• Compare different model architectures or hyperparameters
• Debug training issues (e.g., collapsed loss at certain timesteps)

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Interpolation visualization

• Demonstrate learned latent space structure
• Generate smooth transitions between concepts
• Create visualizations for papers or presentations
• Validate model quality (smooth interpolations = good generalization)
• Compare DDPM vs DDIM sampling quality

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Research applications

• Latent space exploration: Understand what the model learns
• Semantic interpolation: Find meaningful directions in noise space
• Model comparison: Evaluate different training configurations
• Quality metrics: Smooth interpolations correlate with sample quality

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Performance notes

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Computational cost

• per_timestep_loss(): Fast, ~1-2 minutes for 10 batches
• interpolate_noise_and_generate(): Slow, ~8-10 minutes for 8×9 grid (requires 72 full DDPM samplings)
• interpolate_noise_and_generate_ddim(): Same as DDPM, ~8-10 minutes (full 1000 steps)

# In diffusion model, add fast DDIM method:
def sample_ddim_fast(self, x_T, ddim_steps=50):
    # Use only subset of timesteps
    pass

# Then use in interpolation:
def interpolate_noise_fast(diffusion, x_T, ddim_steps=50):
    # Only ~1 minute instead of 10 for same quality
    pass

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Related functions

• DiffusionProcess.add_noise(): Forward diffusion noise addition
• DiffusionProcess.sample(): Standard DDPM sampling
• DiffusionProcess.sample_ddim(): Accelerated DDIM sampling
• See Diffusion process for core diffusion operations