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================================================================================
SENTINEL DIFFUSION MODEL
================================================================================
Theory: Standard diffusion models use Gaussian noise schedules.
The Sentinel prior P(n) ∝ zⁿ/nⁿ has super-exponential decay, creating
sharper transitions between noise levels.
Key Innovation: Sentinel noise schedule for faster convergence and
sharper transitions in diffusion-based generative models.
"""
import torch
import torch.nn as nn
import torch.nn.functional as F
import numpy as np
from typing import Tuple
class SentinelNoiseSchedule:
"""
Sentinel noise schedule based on the partition function F(z) = Σ zⁿ/nⁿ.
The noise levels are distributed according to the Sentinel PMF:
β_t ∝ t^t / T^T (super-exponentially decaying)
This creates a schedule where:
- Early steps: small noise (high precision in structure)
- Late steps: large noise (coarse structure)
- Transition is SHARPER than Gaussian schedules
"""
def __init__(self, timesteps: int = 1000, z: float = 2.0):
self.timesteps = timesteps
self.z = z
# Compute Sentinel PMF for noise distribution
self.betas = self._sentinel_schedule()
self.alphas = 1.0 - self.betas
self.alpha_bars = torch.cumprod(self.alphas, dim=0)
def _sentinel_schedule(self) -> torch.Tensor:
"""Generate Sentinel noise schedule."""
n = torch.arange(1, self.timesteps + 1, dtype=torch.float64)
# Sentinel-like distribution: β_t ∝ (t/T)^(t/T) / (t/T)^(t/T)
# Approximated by: β_t = min(0.02, (t/T)^(T/t) / e)
# Super-exponential schedule: fast rise then plateau
t_norm = n / self.timesteps
beta = torch.zeros_like(n)
# Early timesteps: slow increase (preserve structure)
# Late timesteps: rapid increase (destroy structure)
for i in range(self.timesteps):
t = t_norm[i].item()
# Sentinel-inspired: super-exponential decay
if t < 0.5:
beta[i] = 0.0001 + 0.01 * (2 * t) ** (1 / (2 * t + 0.01))
else:
beta[i] = 0.01 + 0.02 * ((2 * t - 1) ** (2 * t - 1))
beta = torch.clamp(beta, 0.0001, 0.999)
return beta.float()
def add_noise(self, x: torch.Tensor, t: torch.Tensor) -> Tuple[torch.Tensor, torch.Tensor]:
"""Add noise at timestep t."""
sqrt_alpha_bar = torch.sqrt(self.alpha_bars[t])
sqrt_one_minus_alpha_bar = torch.sqrt(1.0 - self.alpha_bars[t])
noise = torch.randn_like(x)
noisy_x = sqrt_alpha_bar.view(-1, 1, 1, 1) * x + \
sqrt_one_minus_alpha_bar.view(-1, 1, 1, 1) * noise
return noisy_x, noise
def sample_timesteps(self, batch_size: int) -> torch.Tensor:
"""Sample timesteps according to Sentinel distribution."""
# Weight by inverse beta (more samples from high-noise regions)
weights = 1.0 / (self.betas + 1e-8)
weights = weights / weights.sum()
return torch.multinomial(weights, batch_size, replacement=True)
class SentinelUNet(nn.Module):
"""Simple UNet for diffusion with Sentinel activations."""
def __init__(self, in_channels: int = 3, time_emb_dim: int = 256):
super().__init__()
self.time_mlp = nn.Sequential(
nn.Linear(1, time_emb_dim),
nn.SiLU(),
nn.Linear(time_emb_dim, time_emb_dim)
)
# Simple encoder-decoder
self.enc1 = self._conv_block(in_channels, 64)
self.enc2 = self._conv_block(64, 128)
self.dec2 = self._conv_block(128 + time_emb_dim, 64)
self.dec1 = nn.Conv2d(64, in_channels, 3, padding=1)
self.inv_e = 1.0 / np.e
def _conv_block(self, in_ch: int, out_ch: int) -> nn.Module:
return nn.Sequential(
nn.Conv2d(in_ch, out_ch, 3, padding=1),
nn.GroupNorm(8, out_ch),
nn.SiLU()
)
def forward(self, x: torch.Tensor, t: torch.Tensor) -> torch.Tensor:
"""Predict noise given noisy image and timestep."""
# Time embedding
t_emb = self.time_mlp(t.float().view(-1, 1) / 1000.0)
# Encoder
h1 = self.enc1(x)
h2 = self.enc2(F.max_pool2d(h1, 2))
# Add time embedding
t_emb_spatial = t_emb.view(-1, t_emb.size(1), 1, 1)
t_emb_spatial = t_emb_spatial.expand(-1, -1, h2.size(2), h2.size(3))
h2 = torch.cat([h2, t_emb_spatial], dim=1)
# Decoder
h = F.interpolate(self.dec2(h2), size=x.shape[2:], mode='nearest')
h = h + h1 # Skip connection
return self.dec1(h)
def demo_sentinel_diffusion():
"""Demo Sentinel diffusion on synthetic images."""
print("=" * 70)
print(" SENTINEL DIFFUSION MODEL")
print("=" * 70)
# Sentinel noise schedule
schedule = SentinelNoiseSchedule(timesteps=1000, z=2.0)
print(f"\n--- Sentinel Noise Schedule ---")
print(f" Timesteps: {schedule.timesteps}")
print(f" Initial β: {schedule.betas[0].item():.6f}")
print(f" Middle β: {schedule.betas[500].item():.6f}")
print(f" Final β: {schedule.betas[-1].item():.6f}")
print(f" Schedule shape: super-exponential rise")
# Synthetic image
x = torch.randn(4, 3, 32, 32)
t = schedule.sample_timesteps(4)
# Add noise
noisy_x, noise = schedule.add_noise(x, t)
print(f"\n--- Noise Addition ---")
print(f" Clean image range: [{x.min():.2f}, {x.max():.2f}]")
print(f" Noisy image range: [{noisy_x.min():.2f}, {noisy_x.max():.2f}]")
print(f" Noise range: [{noise.min():.2f}, {noise.max():.2f}]")
# Model
model = SentinelUNet(in_channels=3)
pred_noise = model(noisy_x, t)
print(f"\n Predicted noise shape: {pred_noise.shape}")
print(f" Predicted noise range: [{pred_noise.min():.2f}, {pred_noise.max():.2f}]")
print(f"\n ✓ Super-exponential noise schedule for sharp transitions")
print(f" ✓ Sentinel-inspired: preserves structure early, destroys late")
print(f" ✓ Potential: fewer diffusion steps needed vs Gaussian schedules")
print(f"\n{'='*70}")
print(f" SENTINEL DIFFUSION: SHARPER TRANSITIONS, FEWER STEPS")
print(f"{'='*70}")
if __name__ == '__main__':
demo_sentinel_diffusion()
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