scheduling_unipc_multistep.py

# Copyright 2025 TSAIL Team and The HuggingFace Team. All rights reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
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#
#     http://www.apache.org/licenses/LICENSE-2.0
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# Unless required by applicable law or agreed to in writing, software
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# DISCLAIMER: check https://huggingface.co/papers/2302.04867 and https://github.com/wl-zhao/UniPC for more info
# The codebase is modified based on https://github.com/huggingface/diffusers/blob/main/src/diffusers/schedulers/scheduling_dpmsolver_multistep.py

import math
from typing import List, Literal, Optional, Tuple, Union

import numpy as np
import torch

from ..configuration_utils import ConfigMixin, register_to_config
from ..utils import deprecate, is_scipy_available
from .scheduling_utils import KarrasDiffusionSchedulers, SchedulerMixin, SchedulerOutput


if is_scipy_available():
    import scipy.stats


# Copied from diffusers.schedulers.scheduling_ddpm.betas_for_alpha_bar
def betas_for_alpha_bar(
    num_diffusion_timesteps: int,
    max_beta: float = 0.999,
    alpha_transform_type: Literal["cosine", "exp"] = "cosine",
) -> torch.Tensor:
    """
    Create a beta schedule that discretizes the given alpha_t_bar function, which defines the cumulative product of
    (1-beta) over time from t = [0,1].

    Contains a function alpha_bar that takes an argument t and transforms it to the cumulative product of (1-beta) up
    to that part of the diffusion process.

    Args:
        num_diffusion_timesteps (`int`):
            The number of betas to produce.
        max_beta (`float`, defaults to `0.999`):
            The maximum beta to use; use values lower than 1 to avoid numerical instability.
        alpha_transform_type (`"cosine"` or `"exp"`, defaults to `"cosine"`):
            The type of noise schedule for `alpha_bar`. Choose from `cosine` or `exp`.

    Returns:
        `torch.Tensor`:
            The betas used by the scheduler to step the model outputs.
    """
    if alpha_transform_type == "cosine":

        def alpha_bar_fn(t):
            return math.cos((t + 0.008) / 1.008 * math.pi / 2) ** 2

    elif alpha_transform_type == "exp":

        def alpha_bar_fn(t):
            return math.exp(t * -12.0)

    else:
        raise ValueError(f"Unsupported alpha_transform_type: {alpha_transform_type}")

    betas = []
    for i in range(num_diffusion_timesteps):
        t1 = i / num_diffusion_timesteps
        t2 = (i + 1) / num_diffusion_timesteps
        betas.append(min(1 - alpha_bar_fn(t2) / alpha_bar_fn(t1), max_beta))
    return torch.tensor(betas, dtype=torch.float32)


# Copied from diffusers.schedulers.scheduling_ddim.rescale_zero_terminal_snr
def rescale_zero_terminal_snr(betas: torch.Tensor) -> torch.Tensor:
    """
    Rescales betas to have zero terminal SNR Based on https://huggingface.co/papers/2305.08891 (Algorithm 1)

    Args:
        betas (`torch.Tensor`):
            The betas that the scheduler is being initialized with.

    Returns:
        `torch.Tensor`:
            Rescaled betas with zero terminal SNR.
    """
    # Convert betas to alphas_bar_sqrt
    alphas = 1.0 - betas
    alphas_cumprod = torch.cumprod(alphas, dim=0)
    alphas_bar_sqrt = alphas_cumprod.sqrt()

    # Store old values.
    alphas_bar_sqrt_0 = alphas_bar_sqrt[0].clone()
    alphas_bar_sqrt_T = alphas_bar_sqrt[-1].clone()

    # Shift so the last timestep is zero.
    alphas_bar_sqrt -= alphas_bar_sqrt_T

    # Scale so the first timestep is back to the old value.
    alphas_bar_sqrt *= alphas_bar_sqrt_0 / (alphas_bar_sqrt_0 - alphas_bar_sqrt_T)

    # Convert alphas_bar_sqrt to betas
    alphas_bar = alphas_bar_sqrt**2  # Revert sqrt
    alphas = alphas_bar[1:] / alphas_bar[:-1]  # Revert cumprod
    alphas = torch.cat([alphas_bar[0:1], alphas])
    betas = 1 - alphas

    return betas


class UniPCMultistepScheduler(SchedulerMixin, ConfigMixin):
    """
    `UniPCMultistepScheduler` is a training-free framework designed for the fast sampling of diffusion models.

    This model inherits from [`SchedulerMixin`] and [`ConfigMixin`]. Check the superclass documentation for the generic
    methods the library implements for all schedulers such as loading and saving.

    Args:
        num_train_timesteps (`int`, defaults to 1000):
            The number of diffusion steps to train the model.
        beta_start (`float`, defaults to 0.0001):
            The starting `beta` value of inference.
        beta_end (`float`, defaults to 0.02):
            The final `beta` value.
        beta_schedule (`"linear"`, `"scaled_linear"`, or `"squaredcos_cap_v2"`, defaults to `"linear"`):
            The beta schedule, a mapping from a beta range to a sequence of betas for stepping the model. Choose from
            `linear`, `scaled_linear`, or `squaredcos_cap_v2`.
        trained_betas (`np.ndarray`, *optional*):
            Pass an array of betas directly to the constructor to bypass `beta_start` and `beta_end`.
        solver_order (`int`, defaults to `2`):
            The UniPC order which can be any positive integer. The effective order of accuracy is `solver_order + 1`
            due to the UniC. It is recommended to use `solver_order=2` for guided sampling, and `solver_order=3` for
            unconditional sampling.
        prediction_type (`"epsilon"`, `"sample"`, `"v_prediction"`, or `"flow_prediction"`, defaults to `"epsilon"`, *optional*):
            Prediction type of the scheduler function; can be `epsilon` (predicts the noise of the diffusion process),
            `sample` (directly predicts the noisy sample`), `v_prediction` (see section 2.4 of [Imagen
            Video](https://huggingface.co/papers/2210.02303) paper), or `flow_prediction`.
        thresholding (`bool`, defaults to `False`):
            Whether to use the "dynamic thresholding" method. This is unsuitable for latent-space diffusion models such
            as Stable Diffusion.
        dynamic_thresholding_ratio (`float`, defaults to 0.995):
            The ratio for the dynamic thresholding method. Valid only when `thresholding=True`.
        sample_max_value (`float`, defaults to 1.0):
            The threshold value for dynamic thresholding. Valid only when `thresholding=True` and `predict_x0=True`.
        predict_x0 (`bool`, defaults to `True`):
            Whether to use the updating algorithm on the predicted x0.
        solver_type (`"bh1"` or `"bh2"`, defaults to `"bh2"`):
            Solver type for UniPC. It is recommended to use `bh1` for unconditional sampling when steps < 10, and `bh2`
            otherwise.
        lower_order_final (`bool`, default `True`):
            Whether to use lower-order solvers in the final steps. Only valid for < 15 inference steps. This can
            stabilize the sampling of DPMSolver for steps < 15, especially for steps <= 10.
        disable_corrector (`list`, default `[]`):
            Decides which step to disable the corrector to mitigate the misalignment between `epsilon_theta(x_t, c)`
            and `epsilon_theta(x_t^c, c)` which can influence convergence for a large guidance scale. Corrector is
            usually disabled during the first few steps.
        solver_p (`SchedulerMixin`, default `None`):
            Any other scheduler that if specified, the algorithm becomes `solver_p + UniC`.
        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.
        use_beta_sigmas (`bool`, *optional*, defaults to `False`):
            Whether to use beta sigmas for step sizes in the noise schedule during the sampling process. Refer to [Beta
            Sampling is All You Need](https://huggingface.co/papers/2407.12173) for more information.
        use_flow_sigmas (`bool`, *optional*, defaults to `False`):
            Whether to use flow sigmas for step sizes in the noise schedule during the sampling process.
        timestep_spacing (`"linspace"`, `"leading"`, or `"trailing"`, 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.
        steps_offset (`int`, defaults to 0):
            An offset added to the inference steps, as required by some model families.
        final_sigmas_type (`"zero"` or `"sigma_min"`, defaults to `"zero"`):
            The final `sigma` value for the noise schedule during the sampling process. If `"sigma_min"`, the final
            sigma is the same as the last sigma in the training schedule. If `zero`, the final sigma is set to 0.
        rescale_betas_zero_snr (`bool`, defaults to `False`):
            Whether to rescale the betas to have zero terminal SNR. This enables the model to generate very bright and
            dark samples instead of limiting it to samples with medium brightness. Loosely related to
            [`--offset_noise`](https://github.com/huggingface/diffusers/blob/74fd735eb073eb1d774b1ab4154a0876eb82f055/examples/dreambooth/train_dreambooth.py#L506).
    """

    _compatibles = [e.name for e in KarrasDiffusionSchedulers]
    order = 1

    @register_to_config
    def __init__(
        self,
        num_train_timesteps: int = 1000,
        beta_start: float = 0.0001,
        beta_end: float = 0.02,
        beta_schedule: Literal["linear", "scaled_linear", "squaredcos_cap_v2"] = "linear",
        trained_betas: Optional[Union[np.ndarray, List[float]]] = None,
        solver_order: int = 2,
        prediction_type: Literal["epsilon", "sample", "v_prediction", "flow_prediction"] = "epsilon",
        thresholding: bool = False,
        dynamic_thresholding_ratio: float = 0.995,
        sample_max_value: float = 1.0,
        predict_x0: bool = True,
        solver_type: Literal["bh1", "bh2"] = "bh2",
        lower_order_final: bool = True,
        disable_corrector: List[int] = [],
        solver_p: Optional[SchedulerMixin] = None,
        use_karras_sigmas: Optional[bool] = False,
        use_exponential_sigmas: Optional[bool] = False,
        use_beta_sigmas: Optional[bool] = False,
        use_flow_sigmas: Optional[bool] = False,
        flow_shift: Optional[float] = 1.0,
        timestep_spacing: Literal["linspace", "leading", "trailing"] = "linspace",
        steps_offset: int = 0,
        final_sigmas_type: Optional[Literal["zero", "sigma_min"]] = "zero",
        rescale_betas_zero_snr: bool = False,
        use_dynamic_shifting: bool = False,
        time_shift_type: Literal["exponential"] = "exponential",
    ) -> None:
        if self.config.use_beta_sigmas and not is_scipy_available():
            raise ImportError("Make sure to install scipy if you want to use beta sigmas.")
        if sum([self.config.use_beta_sigmas, self.config.use_exponential_sigmas, self.config.use_karras_sigmas]) > 1:
            raise ValueError(
                "Only one of `config.use_beta_sigmas`, `config.use_exponential_sigmas`, `config.use_karras_sigmas` can be used."
            )
        if trained_betas is not None:
            self.betas = torch.tensor(trained_betas, dtype=torch.float32)
        elif beta_schedule == "linear":
            self.betas = torch.linspace(beta_start, beta_end, num_train_timesteps, dtype=torch.float32)
        elif beta_schedule == "scaled_linear":
            # this schedule is very specific to the latent diffusion model.
            self.betas = torch.linspace(beta_start**0.5, beta_end**0.5, num_train_timesteps, dtype=torch.float32) ** 2
        elif beta_schedule == "squaredcos_cap_v2":
            # Glide cosine schedule
            self.betas = betas_for_alpha_bar(num_train_timesteps)
        else:
            raise NotImplementedError(f"{beta_schedule} is not implemented for {self.__class__}")

        if rescale_betas_zero_snr:
            self.betas = rescale_zero_terminal_snr(self.betas)

        self.alphas = 1.0 - self.betas
        self.alphas_cumprod = torch.cumprod(self.alphas, dim=0)

        if rescale_betas_zero_snr:
            # Close to 0 without being 0 so first sigma is not inf
            # FP16 smallest positive subnormal works well here
            self.alphas_cumprod[-1] = 2**-24

        # Currently we only support VP-type noise schedule
        self.alpha_t = torch.sqrt(self.alphas_cumprod)
        self.sigma_t = torch.sqrt(1 - self.alphas_cumprod)
        self.lambda_t = torch.log(self.alpha_t) - torch.log(self.sigma_t)
        self.sigmas = ((1 - self.alphas_cumprod) / self.alphas_cumprod) ** 0.5

        # standard deviation of the initial noise distribution
        self.init_noise_sigma = 1.0

        if solver_type not in ["bh1", "bh2"]:
            if solver_type in ["midpoint", "heun", "logrho"]:
                self.register_to_config(solver_type="bh2")
            else:
                raise NotImplementedError(f"{solver_type} is not implemented for {self.__class__}")

        self.predict_x0 = predict_x0
        # setable values
        self.num_inference_steps = None
        timesteps = np.linspace(0, num_train_timesteps - 1, num_train_timesteps, dtype=np.float32)[::-1].copy()
        self.timesteps = torch.from_numpy(timesteps)
        self.model_outputs = [None] * solver_order
        self.timestep_list = [None] * solver_order
        self.lower_order_nums = 0
        self.disable_corrector = disable_corrector
        self.solver_p = solver_p
        self.last_sample = None
        self._step_index = None
        self._begin_index = None
        self.sigmas = self.sigmas.to("cpu")  # to avoid too much CPU/GPU communication

    @property
    def step_index(self) -> Optional[int]:
        """
        The index counter for current timestep. It will increase 1 after each scheduler step.
        """
        return self._step_index

    @property
    def begin_index(self) -> Optional[int]:
        """
        The index for the first timestep. It should be set from pipeline with `set_begin_index` method.
        """
        return self._begin_index

    # Copied from diffusers.schedulers.scheduling_dpmsolver_multistep.DPMSolverMultistepScheduler.set_begin_index
    def set_begin_index(self, begin_index: int = 0) -> None:
        """
        Sets the begin index for the scheduler. This function should be run from pipeline before the inference.

        Args:
            begin_index (`int`, defaults to `0`):
                The begin index for the scheduler.
        """
        self._begin_index = begin_index

    def set_timesteps(
        self, num_inference_steps: int, device: Optional[Union[str, torch.device]] = None, mu: Optional[float] = None
    ) -> None:
        """
        Sets the discrete timesteps used for the diffusion chain (to be run before inference).

        Args:
            num_inference_steps (`int`):
                The number of diffusion steps used when generating samples with a pre-trained model.
            device (`str` or `torch.device`, *optional*):
                The device to which the timesteps should be moved to. If `None`, the timesteps are not moved.
            mu (`float`, *optional*):
                Optional mu parameter for dynamic shifting when using exponential time shift type.
        """
        # "linspace", "leading", "trailing" corresponds to annotation of Table 2. of https://huggingface.co/papers/2305.08891
        if mu is not None:
            assert self.config.use_dynamic_shifting and self.config.time_shift_type == "exponential"
            self.config.flow_shift = np.exp(mu)
        if self.config.timestep_spacing == "linspace":
            timesteps = (
                np.linspace(0, self.config.num_train_timesteps - 1, num_inference_steps + 1)
                .round()[::-1][:-1]
                .copy()
                .astype(np.int64)
            )
        elif self.config.timestep_spacing == "leading":
            step_ratio = self.config.num_train_timesteps // (num_inference_steps + 1)
            # creates integer timesteps by multiplying by ratio
            # casting to int to avoid issues when num_inference_step is power of 3
            timesteps = (np.arange(0, num_inference_steps + 1) * step_ratio).round()[::-1][:-1].copy().astype(np.int64)
            timesteps += self.config.steps_offset
        elif self.config.timestep_spacing == "trailing":
            step_ratio = self.config.num_train_timesteps / num_inference_steps
            # creates integer timesteps by multiplying by ratio
            # casting to int to avoid issues when num_inference_step is power of 3
            timesteps = np.arange(self.config.num_train_timesteps, 0, -step_ratio).round().copy().astype(np.int64)
            timesteps -= 1
        else:
            raise ValueError(
                f"{self.config.timestep_spacing} is not supported. Please make sure to choose one of 'linspace', 'leading' or 'trailing'."
            )

        sigmas = np.array(((1 - self.alphas_cumprod) / self.alphas_cumprod) ** 0.5)
        if self.config.use_karras_sigmas:
            log_sigmas = np.log(sigmas)
            sigmas = np.flip(sigmas).copy()
            sigmas = self._convert_to_karras(in_sigmas=sigmas, num_inference_steps=num_inference_steps)
            timesteps = np.array([self._sigma_to_t(sigma, log_sigmas) for sigma in sigmas]).round()
            if self.config.final_sigmas_type == "sigma_min":
                sigma_last = sigmas[-1]
            elif self.config.final_sigmas_type == "zero":
                sigma_last = 0
            else:
                raise ValueError(
                    f"`final_sigmas_type` must be one of 'zero', or 'sigma_min', but got {self.config.final_sigmas_type}"
                )
            sigmas = np.concatenate([sigmas, [sigma_last]]).astype(np.float32)
        elif self.config.use_exponential_sigmas:
            log_sigmas = np.log(sigmas)
            sigmas = np.flip(sigmas).copy()
            sigmas = self._convert_to_exponential(in_sigmas=sigmas, num_inference_steps=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 = sigmas[-1]
            elif self.config.final_sigmas_type == "zero":
                sigma_last = 0
            else:
                raise ValueError(
                    f"`final_sigmas_type` must be one of 'zero', or 'sigma_min', but got {self.config.final_sigmas_type}"
                )
            sigmas = np.concatenate([sigmas, [sigma_last]]).astype(np.float32)
        elif self.config.use_beta_sigmas:
            log_sigmas = np.log(sigmas)
            sigmas = np.flip(sigmas).copy()
            sigmas = self._convert_to_beta(in_sigmas=sigmas, num_inference_steps=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 = sigmas[-1]
            elif self.config.final_sigmas_type == "zero":
                sigma_last = 0
            else:
                raise ValueError(
                    f"`final_sigmas_type` must be one of 'zero', or 'sigma_min', but got {self.config.final_sigmas_type}"
                )
            sigmas = np.concatenate([sigmas, [sigma_last]]).astype(np.float32)
        elif self.config.use_flow_sigmas:
            alphas = np.linspace(1, 1 / self.config.num_train_timesteps, num_inference_steps + 1)
            sigmas = 1.0 - alphas
            sigmas = np.flip(self.config.flow_shift * sigmas / (1 + (self.config.flow_shift - 1) * sigmas))[:-1].copy()
            timesteps = (sigmas * self.config.num_train_timesteps).copy()
            if self.config.final_sigmas_type == "sigma_min":
                sigma_last = sigmas[-1]
            elif self.config.final_sigmas_type == "zero":
                sigma_last = 0
            else:
                raise ValueError(
                    f"`final_sigmas_type` must be one of 'zero', or 'sigma_min', but got {self.config.final_sigmas_type}"
                )
            sigmas = np.concatenate([sigmas, [sigma_last]]).astype(np.float32)
        else:
            sigmas = np.interp(timesteps, np.arange(0, len(sigmas)), 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":
                sigma_last = 0
            else:
                raise ValueError(
                    f"`final_sigmas_type` must be one of 'zero', or 'sigma_min', but got {self.config.final_sigmas_type}"
                )
            sigmas = np.concatenate([sigmas, [sigma_last]]).astype(np.float32)

        self.sigmas = torch.from_numpy(sigmas)
        self.timesteps = torch.from_numpy(timesteps).to(device=device, dtype=torch.int64)

        self.num_inference_steps = len(timesteps)

        self.model_outputs = [
            None,
        ] * self.config.solver_order
        self.lower_order_nums = 0
        self.last_sample = None
        if self.solver_p:
            self.solver_p.set_timesteps(self.num_inference_steps, device=device)

        # add an index counter for schedulers that allow duplicated timesteps
        self._step_index = None
        self._begin_index = None
        self.sigmas = self.sigmas.to("cpu")  # to avoid too much CPU/GPU communication

    # Copied from diffusers.schedulers.scheduling_ddpm.DDPMScheduler._threshold_sample
    def _threshold_sample(self, sample: torch.Tensor) -> torch.Tensor:
        """
        Apply dynamic thresholding to the predicted sample.

        "Dynamic thresholding: At each sampling step we set s to a certain percentile absolute pixel value in xt0 (the
        prediction of x_0 at timestep t), and if s > 1, then we threshold xt0 to the range [-s, s] and then divide by
        s. Dynamic thresholding pushes saturated pixels (those near -1 and 1) inwards, thereby actively preventing
        pixels from saturation at each step. We find that dynamic thresholding results in significantly better
        photorealism as well as better image-text alignment, especially when using very large guidance weights."

        https://huggingface.co/papers/2205.11487

        Args:
            sample (`torch.Tensor`):
                The predicted sample to be thresholded.

        Returns:
            `torch.Tensor`:
                The thresholded sample.
        """
        dtype = sample.dtype
        batch_size, channels, *remaining_dims = sample.shape

        if dtype not in (torch.float32, torch.float64):
            sample = sample.float()  # upcast for quantile calculation, and clamp not implemented for cpu half

        # Flatten sample for doing quantile calculation along each image
        sample = sample.reshape(batch_size, channels * np.prod(remaining_dims))

        abs_sample = sample.abs()  # "a certain percentile absolute pixel value"

        s = torch.quantile(abs_sample, self.config.dynamic_thresholding_ratio, dim=1)
        s = torch.clamp(
            s, min=1, max=self.config.sample_max_value
        )  # When clamped to min=1, equivalent to standard clipping to [-1, 1]
        s = s.unsqueeze(1)  # (batch_size, 1) because clamp will broadcast along dim=0
        sample = torch.clamp(sample, -s, s) / s  # "we threshold xt0 to the range [-s, s] and then divide by s"

        sample = sample.reshape(batch_size, channels, *remaining_dims)
        sample = sample.to(dtype)

        return sample

    # Copied from diffusers.schedulers.scheduling_euler_discrete.EulerDiscreteScheduler._sigma_to_t
    def _sigma_to_t(self, sigma: np.ndarray, log_sigmas: np.ndarray) -> np.ndarray:
        """
        Convert sigma values to corresponding timestep values through interpolation.

        Args:
            sigma (`np.ndarray`):
                The sigma value(s) to convert to timestep(s).
            log_sigmas (`np.ndarray`):
                The logarithm of the sigma schedule used for interpolation.

        Returns:
            `np.ndarray`:
                The interpolated timestep value(s) corresponding to the input sigma(s).
        """
        # get log sigma
        log_sigma = np.log(np.maximum(sigma, 1e-10))

        # get distribution
        dists = log_sigma - log_sigmas[:, np.newaxis]

        # get sigmas range
        low_idx = np.cumsum((dists >= 0), axis=0).argmax(axis=0).clip(max=log_sigmas.shape[0] - 2)
        high_idx = low_idx + 1

        low = log_sigmas[low_idx]
        high = log_sigmas[high_idx]

        # interpolate sigmas
        w = (low - log_sigma) / (low - high)
        w = np.clip(w, 0, 1)

        # transform interpolation to time range
        t = (1 - w) * low_idx + w * high_idx
        t = t.reshape(sigma.shape)
        return t

    # Copied from diffusers.schedulers.scheduling_dpmsolver_multistep.DPMSolverMultistepScheduler._sigma_to_alpha_sigma_t
    def _sigma_to_alpha_sigma_t(self, sigma: torch.Tensor) -> Tuple[torch.Tensor, torch.Tensor]:
        """
        Convert sigma values to alpha_t and sigma_t values.

        Args:
            sigma (`torch.Tensor`):
                The sigma value(s) to convert.

        Returns:
            `Tuple[torch.Tensor, torch.Tensor]`:
                A tuple containing (alpha_t, sigma_t) values.
        """
        if self.config.use_flow_sigmas:
            alpha_t = 1 - sigma
            sigma_t = sigma
        else:
            alpha_t = 1 / ((sigma**2 + 1) ** 0.5)
            sigma_t = sigma * alpha_t

        return alpha_t, sigma_t

    # Copied from diffusers.schedulers.scheduling_euler_discrete.EulerDiscreteScheduler._convert_to_karras
    def _convert_to_karras(self, in_sigmas: torch.Tensor, num_inference_steps: int) -> torch.Tensor:
        """
        Construct the noise schedule as proposed in [Elucidating the Design Space of Diffusion-Based Generative
        Models](https://huggingface.co/papers/2206.00364).

        Args:
            in_sigmas (`torch.Tensor`):
                The input sigma values to be converted.
            num_inference_steps (`int`):
                The number of inference steps to generate the noise schedule for.

        Returns:
            `torch.Tensor`:
                The converted sigma values following the Karras 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()

        rho = 7.0  # 7.0 is the value used in the paper
        ramp = np.linspace(0, 1, num_inference_steps)
        min_inv_rho = sigma_min ** (1 / rho)
        max_inv_rho = sigma_max ** (1 / rho)
        sigmas = (max_inv_rho + ramp * (min_inv_rho - max_inv_rho)) ** rho
        return sigmas

    # Copied from diffusers.schedulers.scheduling_euler_discrete.EulerDiscreteScheduler._convert_to_exponential
    def _convert_to_exponential(self, in_sigmas: torch.Tensor, num_inference_steps: int) -> torch.Tensor:
        """
        Construct an exponential noise schedule.

        Args:
            in_sigmas (`torch.Tensor`):
                The input sigma values to be converted.
            num_inference_steps (`int`):
                The number of inference steps to generate the noise schedule for.

        Returns:
            `torch.Tensor`:
                The converted sigma values following an exponential 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 = np.exp(np.linspace(math.log(sigma_max), math.log(sigma_min), num_inference_steps))
        return sigmas

    # Copied from diffusers.schedulers.scheduling_euler_discrete.EulerDiscreteScheduler._convert_to_beta
    def _convert_to_beta(
        self, in_sigmas: torch.Tensor, num_inference_steps: int, alpha: float = 0.6, beta: float = 0.6
    ) -> torch.Tensor:
        """
        Construct a beta noise schedule as proposed in [Beta Sampling is All You
        Need](https://huggingface.co/papers/2407.12173).

        Args:
            in_sigmas (`torch.Tensor`):
                The input sigma values to be converted.
            num_inference_steps (`int`):
                The number of inference steps to generate the noise schedule for.
            alpha (`float`, *optional*, defaults to `0.6`):
                The alpha parameter for the beta distribution.
            beta (`float`, *optional*, defaults to `0.6`):
                The beta parameter for the beta distribution.

        Returns:
            `torch.Tensor`:
                The converted sigma values following a beta distribution 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 = np.array(
            [
                sigma_min + (ppf * (sigma_max - sigma_min))
                for ppf in [
                    scipy.stats.beta.ppf(timestep, alpha, beta)
                    for timestep in 1 - np.linspace(0, 1, num_inference_steps)
                ]
            ]
        )
        return sigmas

    def convert_model_output(
        self,
        model_output: torch.Tensor,
        *args,
        sample: torch.Tensor = None,
        **kwargs,
    ) -> torch.Tensor:
        r"""
        Convert the model output to the corresponding type the UniPC algorithm needs.

        Args:
            model_output (`torch.Tensor`):
                The direct output from the learned diffusion model.
            timestep (`int`):
                The current discrete timestep in the diffusion chain.
            sample (`torch.Tensor`):
                A current instance of a sample created by the diffusion process.

        Returns:
            `torch.Tensor`:
                The converted model output.
        """
        timestep = args[0] if len(args) > 0 else kwargs.pop("timestep", None)
        if sample is None:
            if len(args) > 1:
                sample = args[1]
            else:
                raise ValueError("missing `sample` as a required keyword argument")
        if timestep is not None:
            deprecate(
                "timesteps",
                "1.0.0",
                "Passing `timesteps` is deprecated and has no effect as model output conversion is now handled via an internal counter `self.step_index`",
            )

        sigma = self.sigmas[self.step_index]
        alpha_t, sigma_t = self._sigma_to_alpha_sigma_t(sigma)

        if self.predict_x0:
            if self.config.prediction_type == "epsilon":
                x0_pred = (sample - sigma_t * model_output) / alpha_t
            elif self.config.prediction_type == "sample":
                x0_pred = model_output
            elif self.config.prediction_type == "v_prediction":
                x0_pred = alpha_t * sample - sigma_t * model_output
            elif self.config.prediction_type == "flow_prediction":
                sigma_t = self.sigmas[self.step_index]
                x0_pred = sample - sigma_t * model_output
            else:
                raise ValueError(
                    f"prediction_type given as {self.config.prediction_type} must be one of `epsilon`, `sample`, "
                    "`v_prediction`, or `flow_prediction` for the UniPCMultistepScheduler."
                )

            if self.config.thresholding:
                x0_pred = self._threshold_sample(x0_pred)

            return x0_pred
        else:
            if self.config.prediction_type == "epsilon":
                return model_output
            elif self.config.prediction_type == "sample":
                epsilon = (sample - alpha_t * model_output) / sigma_t
                return epsilon
            elif self.config.prediction_type == "v_prediction":
                epsilon = alpha_t * model_output + sigma_t * sample
                return epsilon
            else:
                raise ValueError(
                    f"prediction_type given as {self.config.prediction_type} must be one of `epsilon`, `sample`, or"
                    " `v_prediction` for the UniPCMultistepScheduler."
                )

    def multistep_uni_p_bh_update(
        self,
        model_output: torch.Tensor,
        *args,
        sample: torch.Tensor = None,
        order: int = None,
        **kwargs,
    ) -> torch.Tensor:
        """
        One step for the UniP (B(h) version). Alternatively, `self.solver_p` is used if is specified.

        Args:
            model_output (`torch.Tensor`):
                The direct output from the learned diffusion model at the current timestep.
            prev_timestep (`int`):
                The previous discrete timestep in the diffusion chain.
            sample (`torch.Tensor`):
                A current instance of a sample created by the diffusion process.
            order (`int`):
                The order of UniP at this timestep (corresponds to the *p* in UniPC-p).

        Returns:
            `torch.Tensor`:
                The sample tensor at the previous timestep.
        """
        prev_timestep = args[0] if len(args) > 0 else kwargs.pop("prev_timestep", None)
        if sample is None:
            if len(args) > 1:
                sample = args[1]
            else:
                raise ValueError("missing `sample` as a required keyword argument")
        if order is None:
            if len(args) > 2:
                order = args[2]
            else:
                raise ValueError("missing `order` as a required keyword argument")
        if prev_timestep is not None:
            deprecate(
                "prev_timestep",
                "1.0.0",
                "Passing `prev_timestep` is deprecated and has no effect as model output conversion is now handled via an internal counter `self.step_index`",
            )
        model_output_list = self.model_outputs

        s0 = self.timestep_list[-1]
        m0 = model_output_list[-1]
        x = sample

        if self.solver_p:
            x_t = self.solver_p.step(model_output, s0, x).prev_sample
            return x_t

        sigma_t, sigma_s0 = self.sigmas[self.step_index + 1], self.sigmas[self.step_index]
        alpha_t, sigma_t = self._sigma_to_alpha_sigma_t(sigma_t)
        alpha_s0, sigma_s0 = self._sigma_to_alpha_sigma_t(sigma_s0)

        lambda_t = torch.log(alpha_t) - torch.log(sigma_t)
        lambda_s0 = torch.log(alpha_s0) - torch.log(sigma_s0)

        h = lambda_t - lambda_s0
        device = sample.device

        rks = []
        D1s = []
        for i in range(1, order):
            si = self.step_index - i
            mi = model_output_list[-(i + 1)]
            alpha_si, sigma_si = self._sigma_to_alpha_sigma_t(self.sigmas[si])
            lambda_si = torch.log(alpha_si) - torch.log(sigma_si)
            rk = (lambda_si - lambda_s0) / h
            rks.append(rk)
            D1s.append((mi - m0) / rk)

        rks.append(1.0)
        rks = torch.tensor(rks, device=device)

        R = []
        b = []

        hh = -h if self.predict_x0 else h
        h_phi_1 = torch.expm1(hh)  # h\phi_1(h) = e^h - 1
        h_phi_k = h_phi_1 / hh - 1

        factorial_i = 1

        if self.config.solver_type == "bh1":
            B_h = hh
        elif self.config.solver_type == "bh2":
            B_h = torch.expm1(hh)
        else:
            raise NotImplementedError()

        for i in range(1, order + 1):
            R.append(torch.pow(rks, i - 1))
            b.append(h_phi_k * factorial_i / B_h)
            factorial_i *= i + 1
            h_phi_k = h_phi_k / hh - 1 / factorial_i

        R = torch.stack(R)
        b = torch.tensor(b, device=device)

        if len(D1s) > 0:
            D1s = torch.stack(D1s, dim=1)  # (B, K)
            # for order 2, we use a simplified version
            if order == 2:
                rhos_p = torch.tensor([0.5], dtype=x.dtype, device=device)
            else:
                rhos_p = torch.linalg.solve(R[:-1, :-1], b[:-1]).to(device).to(x.dtype)
        else:
            D1s = None

        if self.predict_x0:
            x_t_ = sigma_t / sigma_s0 * x - alpha_t * h_phi_1 * m0
            if D1s is not None:
                pred_res = torch.einsum("k,bkc...->bc...", rhos_p, D1s)
            else:
                pred_res = 0
            x_t = x_t_ - alpha_t * B_h * pred_res
        else:
            x_t_ = alpha_t / alpha_s0 * x - sigma_t * h_phi_1 * m0
            if D1s is not None:
                pred_res = torch.einsum("k,bkc...->bc...", rhos_p, D1s)
            else:
                pred_res = 0
            x_t = x_t_ - sigma_t * B_h * pred_res

        x_t = x_t.to(x.dtype)
        return x_t

    def multistep_uni_c_bh_update(
        self,
        this_model_output: torch.Tensor,
        *args,
        last_sample: torch.Tensor = None,
        this_sample: torch.Tensor = None,
        order: int = None,
        **kwargs,
    ) -> torch.Tensor:
        """
        One step for the UniC (B(h) version).

        Args:
            this_model_output (`torch.Tensor`):
                The model outputs at `x_t`.
            this_timestep (`int`):
                The current timestep `t`.
            last_sample (`torch.Tensor`):
                The generated sample before the last predictor `x_{t-1}`.
            this_sample (`torch.Tensor`):
                The generated sample after the last predictor `x_{t}`.
            order (`int`):
                The `p` of UniC-p at this step. The effective order of accuracy should be `order + 1`.

        Returns:
            `torch.Tensor`:
                The corrected sample tensor at the current timestep.
        """
        this_timestep = args[0] if len(args) > 0 else kwargs.pop("this_timestep", None)
        if last_sample is None:
            if len(args) > 1:
                last_sample = args[1]
            else:
                raise ValueError("missing `last_sample` as a required keyword argument")
        if this_sample is None:
            if len(args) > 2:
                this_sample = args[2]
            else:
                raise ValueError("missing `this_sample` as a required keyword argument")
        if order is None:
            if len(args) > 3:
                order = args[3]
            else:
                raise ValueError("missing `order` as a required keyword argument")
        if this_timestep is not None:
            deprecate(
                "this_timestep",
                "1.0.0",
                "Passing `this_timestep` is deprecated and has no effect as model output conversion is now handled via an internal counter `self.step_index`",
            )

        model_output_list = self.model_outputs

        m0 = model_output_list[-1]
        x = last_sample
        x_t = this_sample
        model_t = this_model_output

        sigma_t, sigma_s0 = self.sigmas[self.step_index], self.sigmas[self.step_index - 1]
        alpha_t, sigma_t = self._sigma_to_alpha_sigma_t(sigma_t)
        alpha_s0, sigma_s0 = self._sigma_to_alpha_sigma_t(sigma_s0)

        lambda_t = torch.log(alpha_t) - torch.log(sigma_t)
        lambda_s0 = torch.log(alpha_s0) - torch.log(sigma_s0)

        h = lambda_t - lambda_s0
        device = this_sample.device

        rks = []
        D1s = []
        for i in range(1, order):
            si = self.step_index - (i + 1)
            mi = model_output_list[-(i + 1)]
            alpha_si, sigma_si = self._sigma_to_alpha_sigma_t(self.sigmas[si])
            lambda_si = torch.log(alpha_si) - torch.log(sigma_si)
            rk = (lambda_si - lambda_s0) / h
            rks.append(rk)
            D1s.append((mi - m0) / rk)

        rks.append(1.0)
        rks = torch.tensor(rks, device=device)

        R = []
        b = []

        hh = -h if self.predict_x0 else h
        h_phi_1 = torch.expm1(hh)  # h\phi_1(h) = e^h - 1
        h_phi_k = h_phi_1 / hh - 1

        factorial_i = 1

        if self.config.solver_type == "bh1":
            B_h = hh
        elif self.config.solver_type == "bh2":
            B_h = torch.expm1(hh)
        else:
            raise NotImplementedError()

        for i in range(1, order + 1):
            R.append(torch.pow(rks, i - 1))
            b.append(h_phi_k * factorial_i / B_h)
            factorial_i *= i + 1
            h_phi_k = h_phi_k / hh - 1 / factorial_i

        R = torch.stack(R)
        b = torch.tensor(b, device=device)

        if len(D1s) > 0:
            D1s = torch.stack(D1s, dim=1)
        else:
            D1s = None

        # for order 1, we use a simplified version
        if order == 1:
            rhos_c = torch.tensor([0.5], dtype=x.dtype, device=device)
        else:
            rhos_c = torch.linalg.solve(R, b).to(device).to(x.dtype)

        if self.predict_x0:
            x_t_ = sigma_t / sigma_s0 * x - alpha_t * h_phi_1 * m0
            if D1s is not None:
                corr_res = torch.einsum("k,bkc...->bc...", rhos_c[:-1], D1s)
            else:
                corr_res = 0
            D1_t = model_t - m0
            x_t = x_t_ - alpha_t * B_h * (corr_res + rhos_c[-1] * D1_t)
        else:
            x_t_ = alpha_t / alpha_s0 * x - sigma_t * h_phi_1 * m0
            if D1s is not None:
                corr_res = torch.einsum("k,bkc...->bc...", rhos_c[:-1], D1s)
            else:
                corr_res = 0
            D1_t = model_t - m0
            x_t = x_t_ - sigma_t * B_h * (corr_res + rhos_c[-1] * D1_t)
        x_t = x_t.to(x.dtype)
        return x_t

    # Copied from diffusers.schedulers.scheduling_dpmsolver_multistep.DPMSolverMultistepScheduler.index_for_timestep
    def index_for_timestep(
        self, timestep: Union[int, torch.Tensor], schedule_timesteps: Optional[torch.Tensor] = None
    ) -> int:
        """
        Find the index for a given timestep in the schedule.

        Args:
            timestep (`int` or `torch.Tensor`):
                The timestep for which to find the index.
            schedule_timesteps (`torch.Tensor`, *optional*):
                The timestep schedule to search in. If `None`, uses `self.timesteps`.

        Returns:
            `int`:
                The index of the timestep in the schedule.
        """
        if schedule_timesteps is None:
            schedule_timesteps = self.timesteps

        index_candidates = (schedule_timesteps == timestep).nonzero()

        if len(index_candidates) == 0:
            step_index = len(self.timesteps) - 1
        # The sigma index that is taken for the **very** first `step`
        # is always the second index (or the last index if there is only 1)
        # This way we can ensure we don't accidentally skip a sigma in
        # case we start in the middle of the denoising schedule (e.g. for image-to-image)
        elif len(index_candidates) > 1:
            step_index = index_candidates[1].item()
        else:
            step_index = index_candidates[0].item()

        return step_index

    # Copied from diffusers.schedulers.scheduling_dpmsolver_multistep.DPMSolverMultistepScheduler._init_step_index
    def _init_step_index(self, timestep: Union[int, torch.Tensor]) -> None:
        """
        Initialize the step_index counter for the scheduler.

        Args:
            timestep (`int` or `torch.Tensor`):
                The current timestep for which to initialize the step index.
        """

        if self.begin_index is None:
            if isinstance(timestep, torch.Tensor):
                timestep = timestep.to(self.timesteps.device)
            self._step_index = self.index_for_timestep(timestep)
        else:
            self._step_index = self._begin_index

    def step(
        self,
        model_output: torch.Tensor,
        timestep: Union[int, torch.Tensor],
        sample: torch.Tensor,
        return_dict: bool = True,
    ) -> Union[SchedulerOutput, Tuple]:
        """
        Predict the sample from the previous timestep by reversing the SDE. This function propagates the sample with
        the multistep UniPC.

        Args:
            model_output (`torch.Tensor`):
                The direct output from learned diffusion model.
            timestep (`int` or `torch.Tensor`):
                The current discrete timestep in the diffusion chain.
            sample (`torch.Tensor`):
                A current instance of a sample created by the diffusion process.
            return_dict (`bool`, defaults to `True`):
                Whether or not to return a [`~schedulers.scheduling_utils.SchedulerOutput`] or `tuple`.

        Returns:
            [`~schedulers.scheduling_utils.SchedulerOutput`] or `tuple`:
                If return_dict is `True`, [`~schedulers.scheduling_utils.SchedulerOutput`] is returned, otherwise a
                tuple is returned where the first element is the sample tensor.

        """
        if self.num_inference_steps is None:
            raise ValueError(
                "Number of inference steps is 'None', you need to run 'set_timesteps' after creating the scheduler"
            )

        if self.step_index is None:
            self._init_step_index(timestep)

        use_corrector = (
            self.step_index > 0 and self.step_index - 1 not in self.disable_corrector and self.last_sample is not None
        )

        model_output_convert = self.convert_model_output(model_output, sample=sample)
        if use_corrector:
            sample = self.multistep_uni_c_bh_update(
                this_model_output=model_output_convert,
                last_sample=self.last_sample,
                this_sample=sample,
                order=self.this_order,
            )

        for i in range(self.config.solver_order - 1):
            self.model_outputs[i] = self.model_outputs[i + 1]
            self.timestep_list[i] = self.timestep_list[i + 1]

        self.model_outputs[-1] = model_output_convert
        self.timestep_list[-1] = timestep

        if self.config.lower_order_final:
            this_order = min(self.config.solver_order, len(self.timesteps) - self.step_index)
        else:
            this_order = self.config.solver_order

        self.this_order = min(this_order, self.lower_order_nums + 1)  # warmup for multistep
        assert self.this_order > 0

        self.last_sample = sample
        prev_sample = self.multistep_uni_p_bh_update(
            model_output=model_output,  # pass the original non-converted model output, in case solver-p is used
            sample=sample,
            order=self.this_order,
        )

        if self.lower_order_nums < self.config.solver_order:
            self.lower_order_nums += 1

        # upon completion increase step index by one
        self._step_index += 1

        if not return_dict:
            return (prev_sample,)

        return SchedulerOutput(prev_sample=prev_sample)

    def scale_model_input(self, sample: torch.Tensor, *args, **kwargs) -> torch.Tensor:
        """
        Ensures interchangeability with schedulers that need to scale the denoising model input depending on the
        current timestep.

        Args:
            sample (`torch.Tensor`):
                The input sample.

        Returns:
            `torch.Tensor`:
                A scaled input sample.
        """
        return sample

    # Copied from diffusers.schedulers.scheduling_dpmsolver_multistep.DPMSolverMultistepScheduler.add_noise
    def add_noise(
        self,
        original_samples: torch.Tensor,
        noise: torch.Tensor,
        timesteps: torch.IntTensor,
    ) -> torch.Tensor:
        """
        Add noise to the original samples according to the noise schedule at the specified timesteps.

        Args:
            original_samples (`torch.Tensor`):
                The original samples without noise.
            noise (`torch.Tensor`):
                The noise to add to the samples.
            timesteps (`torch.IntTensor`):
                The timesteps at which to add noise to the samples.

        Returns:
            `torch.Tensor`:
                The noisy samples.
        """
        # Make sure sigmas and timesteps have the same device and dtype as original_samples
        sigmas = self.sigmas.to(device=original_samples.device, dtype=original_samples.dtype)
        if original_samples.device.type == "mps" and torch.is_floating_point(timesteps):
            # mps does not support float64
            schedule_timesteps = self.timesteps.to(original_samples.device, dtype=torch.float32)
            timesteps = timesteps.to(original_samples.device, dtype=torch.float32)
        else:
            schedule_timesteps = self.timesteps.to(original_samples.device)
            timesteps = timesteps.to(original_samples.device)

        # begin_index is None when the scheduler is used for training or pipeline does not implement set_begin_index
        if self.begin_index is None:
            step_indices = [self.index_for_timestep(t, schedule_timesteps) for t in timesteps]
        elif self.step_index is not None:
            # add_noise is called after first denoising step (for inpainting)
            step_indices = [self.step_index] * timesteps.shape[0]
        else:
            # add noise is called before first denoising step to create initial latent(img2img)
            step_indices = [self.begin_index] * timesteps.shape[0]

        sigma = sigmas[step_indices].flatten()
        while len(sigma.shape) < len(original_samples.shape):
            sigma = sigma.unsqueeze(-1)

        alpha_t, sigma_t = self._sigma_to_alpha_sigma_t(sigma)
        noisy_samples = alpha_t * original_samples + sigma_t * noise
        return noisy_samples

    def __len__(self) -> int:
        return self.config.num_train_timesteps