[Schedulers] Add exponential sigmas / exponential noise schedule (#9499)

* exponential sigmas

* Apply suggestions from code review
Co-authored-by: YiYi Xu <yixu310@gmail.com>

* make style

---------
Co-authored-by: YiYi Xu <yixu310@gmail.com>

src/diffusers/schedulers/scheduling_euler_discrete.py

@@ -158,6 +158,8 @@ class EulerDiscreteScheduler(SchedulerMixin, ConfigMixin):
         use_karras_sigmas (`bool`, *optional*, defaults to `False`):
             Whether to use Karras sigmas for step sizes in the noise schedule during the sampling process. If `True`,
             the sigmas are determined according to a sequence of noise levels {σi}.
+        use_exponential_sigmas (`bool`, *optional*, defaults to `False`):
+            Whether to use exponential sigmas for step sizes in the noise schedule during the sampling process.
         timestep_spacing (`str`, defaults to `"linspace"`):
             The way the timesteps should be scaled. Refer to Table 2 of the [Common Diffusion Noise Schedules and
             Sample Steps are Flawed](https://huggingface.co/papers/2305.08891) for more information.
@@ -186,6 +188,7 @@ class EulerDiscreteScheduler(SchedulerMixin, ConfigMixin):
         prediction_type: str = "epsilon",
         interpolation_type: str = "linear",
         use_karras_sigmas: Optional[bool] = False,
+        use_exponential_sigmas: Optional[bool] = False,
         sigma_min: Optional[float] = None,
         sigma_max: Optional[float] = None,
         timestep_spacing: str = "linspace",
@@ -235,6 +238,7 @@ class EulerDiscreteScheduler(SchedulerMixin, ConfigMixin):
 
         self.is_scale_input_called = False
         self.use_karras_sigmas = use_karras_sigmas
+        self.use_exponential_sigmas = use_exponential_sigmas
 
         self._step_index = None
         self._begin_index = None
@@ -332,6 +336,12 @@ class EulerDiscreteScheduler(SchedulerMixin, ConfigMixin):
             raise ValueError("Can only pass one of `num_inference_steps` or `timesteps` or `sigmas`.")
         if timesteps is not None and self.config.use_karras_sigmas:
             raise ValueError("Cannot set `timesteps` with `config.use_karras_sigmas = True`.")
+        if timesteps is not None and self.config.use_exponential_sigmas:
+            raise ValueError("Cannot set `timesteps` with `config.use_exponential_sigmas = True`.")
+        if self.config.use_exponential_sigmas and self.config.use_karras_sigmas:
+            raise ValueError(
+                "Cannot set both `config.use_exponential_sigmas = True` and config.use_karras_sigmas = True`"
+            )
         if (
             timesteps is not None
             and self.config.timestep_type == "continuous"
@@ -396,6 +406,10 @@ class EulerDiscreteScheduler(SchedulerMixin, ConfigMixin):
                 sigmas = self._convert_to_karras(in_sigmas=sigmas, num_inference_steps=self.num_inference_steps)
                 timesteps = np.array([self._sigma_to_t(sigma, log_sigmas) for sigma in sigmas])
 
+            elif self.config.use_exponential_sigmas:
+                sigmas = self._convert_to_exponential(in_sigmas=sigmas, num_inference_steps=self.num_inference_steps)
+                timesteps = np.array([self._sigma_to_t(sigma, log_sigmas) for sigma in sigmas])
+
             if self.config.final_sigmas_type == "sigma_min":
                 sigma_last = ((1 - self.alphas_cumprod[0]) / self.alphas_cumprod[0]) ** 0.5
             elif self.config.final_sigmas_type == "zero":
@@ -468,6 +482,28 @@ class EulerDiscreteScheduler(SchedulerMixin, ConfigMixin):
         sigmas = (max_inv_rho + ramp * (min_inv_rho - max_inv_rho)) ** rho
         return sigmas
 
+    # Copied from https://github.com/crowsonkb/k-diffusion/blob/686dbad0f39640ea25c8a8c6a6e56bb40eacefa2/k_diffusion/sampling.py#L26
+    def _convert_to_exponential(self, in_sigmas: torch.Tensor, num_inference_steps: int) -> torch.Tensor:
+        """Constructs an exponential noise schedule."""
+
+        # Hack to make sure that other schedulers which copy this function don't break
+        # TODO: Add this logic to the other schedulers
+        if hasattr(self.config, "sigma_min"):
+            sigma_min = self.config.sigma_min
+        else:
+            sigma_min = None
+
+        if hasattr(self.config, "sigma_max"):
+            sigma_max = self.config.sigma_max
+        else:
+            sigma_max = None
+
+        sigma_min = sigma_min if sigma_min is not None else in_sigmas[-1].item()
+        sigma_max = sigma_max if sigma_max is not None else in_sigmas[0].item()
+
+        sigmas = torch.linspace(math.log(sigma_max), math.log(sigma_min), num_inference_steps).exp()
+        return sigmas
+
     def index_for_timestep(self, timestep, schedule_timesteps=None):
         if schedule_timesteps is None:
             schedule_timesteps = self.timesteps