improve ddim comments

models/vision/ddim/modeling_ddim.py

@@ -34,49 +34,68 @@ class DDIM(DiffusionPipeline):
         inference_step_times = range(0, num_trained_timesteps, num_trained_timesteps // num_inference_steps)
 
         self.unet.to(torch_device)
+
+        # Sample gaussian noise to begin loop
         image = self.noise_scheduler.sample_noise(
             (batch_size, self.unet.in_channels, self.unet.resolution, self.unet.resolution),
             device=torch_device,
             generator=generator,
         )
 
+        # See formulas (9), (10) and (7) of DDIM paper https://arxiv.org/pdf/2010.02502.pdf
+        # Ideally, read DDIM paper in-detail understanding
+
+        # Notation (<variable name> -> <name in paper>
+        # - pred_noise_t -> e_theta(x_t, t)
+        # - pred_original_image -> f_theta(x_t, t) or x_0
+        # - std_dev_t -> sigma_t
+        # - eta -> η
+        # - pred_image_direction -> "direction pointingc to x_t"
+        # - pred_prev_image -> "x_t-1"
         for t in tqdm.tqdm(reversed(range(num_inference_steps)), total=num_inference_steps):
-            # get actual t and t-1
+            # 1. predict noise residual
+            with torch.no_grad():
+                pred_noise_t = self.unet(image, inference_step_times[t])
+
+            # 2. get actual t and t-1
             train_step = inference_step_times[t]
             prev_train_step = inference_step_times[t - 1] if t > 0 else -1
 
-            # compute alphas
+            # 3. compute alphas, betas
             alpha_prod_t = self.noise_scheduler.get_alpha_prod(train_step)
             alpha_prod_t_prev = self.noise_scheduler.get_alpha_prod(prev_train_step)
-            alpha_prod_t_rsqrt = 1 / alpha_prod_t.sqrt()
-            alpha_prod_t_prev_rsqrt = 1 / alpha_prod_t_prev.sqrt()
-            beta_prod_t_sqrt = (1 - alpha_prod_t).sqrt()
-            beta_prod_t_prev_sqrt = (1 - alpha_prod_t_prev).sqrt()
-
-            # compute relevant coefficients
-            coeff_1 = (
-                (alpha_prod_t_prev - alpha_prod_t).sqrt()
-                * alpha_prod_t_prev_rsqrt
-                * beta_prod_t_prev_sqrt
-                / beta_prod_t_sqrt
-                * eta
-            )
-            coeff_2 = ((1 - alpha_prod_t_prev) - coeff_1**2).sqrt()
+            beta_prod_t = (1 - alpha_prod_t)
+            beta_prod_t_prev = (1 - alpha_prod_t_prev)
 
-            # model forward
-            with torch.no_grad():
-                noise_residual = self.unet(image, train_step)
+            # 4. Compute predicted previous image from predicted noise
 
-            # predict mean of prev image
-            pred_mean = alpha_prod_t_rsqrt * (image - beta_prod_t_sqrt * noise_residual)
-            pred_mean = torch.clamp(pred_mean, -1, 1)
-            pred_mean = (1 / alpha_prod_t_prev_rsqrt) * pred_mean + coeff_2 * noise_residual
+            # First: compute predicted original image from predicted noise also called
+            # "predicted x_0" of formula (12) from https://arxiv.org/pdf/2010.02502.pdf
+            pred_original_image = (image - beta_prod_t.sqrt() * pred_noise_t) / alpha_prod_t.sqrt()
 
-            # if eta > 0.0 add noise. Note eta = 1.0 essentially corresponds to DDPM
+            # Second: Clip "predicted x_0"
+            pred_original_image = torch.clamp(pred_original_image, -1, 1)
+
+            # Third: Compute variance: "sigma_t(η)" -> see formula (16)
+            # σ_t = sqrt((1 − α_t−1)/(1 − α_t)) * sqrt(1 − α_t/α_t−1)
+            std_dev_t = (beta_prod_t_prev / beta_prod_t).sqrt() * (1 - alpha_prod_t / alpha_prod_t_prev).sqrt()
+            std_dev_t = eta * std_dev_t
+
+            # Fourth: Compute "direction pointing to x_t" of formula (12) from https://arxiv.org/pdf/2010.02502.pdf
+            pred_image_direction = (1 - alpha_prod_t_prev - std_dev_t**2).sqrt() * pred_noise_t
+
+            # Fifth: Compute x_t without "random noise" of formula (12) from https://arxiv.org/pdf/2010.02502.pdf
+            pred_prev_image = alpha_prod_t_prev.sqrt() * pred_original_image + pred_image_direction
+
+            # 5. Sample x_t-1 image optionally if η > 0.0 by adding noise to pred_prev_image
+            # Note: eta = 1.0 essentially corresponds to DDPM
             if eta > 0.0:
                 noise = self.noise_scheduler.sample_noise(image.shape, device=image.device, generator=generator)
-                image = pred_mean + coeff_1 * noise
+                prev_image = pred_prev_image + std_dev_t * noise
             else:
-                image = pred_mean
+                prev_image = pred_prev_image
+
+            # 6. Set current image to prev_image: x_t -> x_t-1
+            image = prev_image
 
         return image