1307 lines
67 KiB
Python
1307 lines
67 KiB
Python
import torch
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class NoiseScheduleVP:
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def __init__(
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self,
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schedule='discrete',
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betas=None,
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alphas_cumprod=None,
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continuous_beta_0=0.1,
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continuous_beta_1=20.,
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dtype=torch.float32,
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):
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"""Create a wrapper class for the forward SDE (VP type).
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***
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Update: We support discrete-time diffusion models by implementing a picewise linear interpolation for log_alpha_t.
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We recommend to use schedule='discrete' for the discrete-time diffusion models, especially for high-resolution images.
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***
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The forward SDE ensures that the condition distribution q_{t|0}(x_t | x_0) = N ( alpha_t * x_0, sigma_t^2 * I ).
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We further define lambda_t = log(alpha_t) - log(sigma_t), which is the half-logSNR (described in the DPM-Solver paper).
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Therefore, we implement the functions for computing alpha_t, sigma_t and lambda_t. For t in [0, T], we have:
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log_alpha_t = self.marginal_log_mean_coeff(t)
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sigma_t = self.marginal_std(t)
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lambda_t = self.marginal_lambda(t)
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Moreover, as lambda(t) is an invertible function, we also support its inverse function:
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t = self.inverse_lambda(lambda_t)
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===============================================================
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We support both discrete-time DPMs (trained on n = 0, 1, ..., N-1) and continuous-time DPMs (trained on t in [t_0, T]).
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1. For discrete-time DPMs:
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For discrete-time DPMs trained on n = 0, 1, ..., N-1, we convert the discrete steps to continuous time steps by:
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t_i = (i + 1) / N
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e.g. for N = 1000, we have t_0 = 1e-3 and T = t_{N-1} = 1.
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We solve the corresponding diffusion ODE from time T = 1 to time t_0 = 1e-3.
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Args:
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betas: A `torch.Tensor`. The beta array for the discrete-time DPM. (See the original DDPM paper for details)
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alphas_cumprod: A `torch.Tensor`. The cumprod alphas for the discrete-time DPM. (See the original DDPM paper for details)
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Note that we always have alphas_cumprod = cumprod(1 - betas). Therefore, we only need to set one of `betas` and `alphas_cumprod`.
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**Important**: Please pay special attention for the args for `alphas_cumprod`:
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The `alphas_cumprod` is the \hat{alpha_n} arrays in the notations of DDPM. Specifically, DDPMs assume that
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q_{t_n | 0}(x_{t_n} | x_0) = N ( \sqrt{\hat{alpha_n}} * x_0, (1 - \hat{alpha_n}) * I ).
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Therefore, the notation \hat{alpha_n} is different from the notation alpha_t in DPM-Solver. In fact, we have
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alpha_{t_n} = \sqrt{\hat{alpha_n}},
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and
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log(alpha_{t_n}) = 0.5 * log(\hat{alpha_n}).
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2. For continuous-time DPMs:
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We support the linear VPSDE for the continuous time setting. The hyperparameters for the noise
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schedule are the default settings in Yang Song's ScoreSDE:
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Args:
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beta_min: A `float` number. The smallest beta for the linear schedule.
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beta_max: A `float` number. The largest beta for the linear schedule.
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T: A `float` number. The ending time of the forward process.
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===============================================================
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Args:
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schedule: A `str`. The noise schedule of the forward SDE. 'discrete' for discrete-time DPMs,
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'linear' for continuous-time DPMs.
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Returns:
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A wrapper object of the forward SDE (VP type).
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===============================================================
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Example:
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# For discrete-time DPMs, given betas (the beta array for n = 0, 1, ..., N - 1):
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>>> ns = NoiseScheduleVP('discrete', betas=betas)
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# For discrete-time DPMs, given alphas_cumprod (the \hat{alpha_n} array for n = 0, 1, ..., N - 1):
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>>> ns = NoiseScheduleVP('discrete', alphas_cumprod=alphas_cumprod)
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# For continuous-time DPMs (VPSDE), linear schedule:
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>>> ns = NoiseScheduleVP('linear', continuous_beta_0=0.1, continuous_beta_1=20.)
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"""
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if schedule not in ['discrete', 'linear']:
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raise ValueError("Unsupported noise schedule {}. The schedule needs to be 'discrete' or 'linear'".format(schedule))
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self.schedule = schedule
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if schedule == 'discrete':
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if betas is not None:
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log_alphas = 0.5 * torch.log(1 - betas).cumsum(dim=0)
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else:
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assert alphas_cumprod is not None
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log_alphas = 0.5 * torch.log(alphas_cumprod)
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self.T = 1.
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self.log_alpha_array = self.numerical_clip_alpha(log_alphas).reshape((1, -1,)).to(dtype=dtype)
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self.total_N = self.log_alpha_array.shape[1]
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self.t_array = torch.linspace(0., 1., self.total_N + 1)[1:].reshape((1, -1)).to(dtype=dtype)
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else:
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self.T = 1.
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self.total_N = 1000
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self.beta_0 = continuous_beta_0
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self.beta_1 = continuous_beta_1
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def numerical_clip_alpha(self, log_alphas, clipped_lambda=-5.1):
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"""
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For some beta schedules such as cosine schedule, the log-SNR has numerical isssues.
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We clip the log-SNR near t=T within -5.1 to ensure the stability.
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Such a trick is very useful for diffusion models with the cosine schedule, such as i-DDPM, guided-diffusion and GLIDE.
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"""
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log_sigmas = 0.5 * torch.log(1. - torch.exp(2. * log_alphas))
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lambs = log_alphas - log_sigmas
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idx = torch.searchsorted(torch.flip(lambs, [0]), clipped_lambda)
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if idx > 0:
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log_alphas = log_alphas[:-idx]
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return log_alphas
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def marginal_log_mean_coeff(self, t):
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"""
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Compute log(alpha_t) of a given continuous-time label t in [0, T].
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"""
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if self.schedule == 'discrete':
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return interpolate_fn(t.reshape((-1, 1)), self.t_array.to(t.device), self.log_alpha_array.to(t.device)).reshape((-1))
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elif self.schedule == 'linear':
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return -0.25 * t ** 2 * (self.beta_1 - self.beta_0) - 0.5 * t * self.beta_0
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def marginal_alpha(self, t):
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"""
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Compute alpha_t of a given continuous-time label t in [0, T].
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"""
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return torch.exp(self.marginal_log_mean_coeff(t))
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def marginal_std(self, t):
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"""
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Compute sigma_t of a given continuous-time label t in [0, T].
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"""
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return torch.sqrt(1. - torch.exp(2. * self.marginal_log_mean_coeff(t)))
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def marginal_lambda(self, t):
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"""
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Compute lambda_t = log(alpha_t) - log(sigma_t) of a given continuous-time label t in [0, T].
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"""
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log_mean_coeff = self.marginal_log_mean_coeff(t)
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log_std = 0.5 * torch.log(1. - torch.exp(2. * log_mean_coeff))
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return log_mean_coeff - log_std
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def inverse_lambda(self, lamb):
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"""
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Compute the continuous-time label t in [0, T] of a given half-logSNR lambda_t.
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"""
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if self.schedule == 'linear':
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tmp = 2. * (self.beta_1 - self.beta_0) * torch.logaddexp(-2. * lamb, torch.zeros((1,)).to(lamb))
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Delta = self.beta_0**2 + tmp
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return tmp / (torch.sqrt(Delta) + self.beta_0) / (self.beta_1 - self.beta_0)
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elif self.schedule == 'discrete':
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log_alpha = -0.5 * torch.logaddexp(torch.zeros((1,)).to(lamb.device), -2. * lamb)
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t = interpolate_fn(log_alpha.reshape((-1, 1)), torch.flip(self.log_alpha_array.to(lamb.device), [1]), torch.flip(self.t_array.to(lamb.device), [1]))
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return t.reshape((-1,))
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def model_wrapper(
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model,
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noise_schedule,
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model_type="noise",
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model_kwargs={},
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guidance_type="uncond",
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condition=None,
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unconditional_condition=None,
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guidance_scale=1.,
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classifier_fn=None,
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classifier_kwargs={},
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):
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"""Create a wrapper function for the noise prediction model.
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DPM-Solver needs to solve the continuous-time diffusion ODEs. For DPMs trained on discrete-time labels, we need to
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firstly wrap the model function to a noise prediction model that accepts the continuous time as the input.
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We support four types of the diffusion model by setting `model_type`:
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1. "noise": noise prediction model. (Trained by predicting noise).
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2. "x_start": data prediction model. (Trained by predicting the data x_0 at time 0).
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3. "v": velocity prediction model. (Trained by predicting the velocity).
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The "v" prediction is derivation detailed in Appendix D of [1], and is used in Imagen-Video [2].
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[1] Salimans, Tim, and Jonathan Ho. "Progressive distillation for fast sampling of diffusion models."
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arXiv preprint arXiv:2202.00512 (2022).
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[2] Ho, Jonathan, et al. "Imagen Video: High Definition Video Generation with Diffusion Models."
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arXiv preprint arXiv:2210.02303 (2022).
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4. "score": marginal score function. (Trained by denoising score matching).
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Note that the score function and the noise prediction model follows a simple relationship:
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```
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noise(x_t, t) = -sigma_t * score(x_t, t)
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```
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We support three types of guided sampling by DPMs by setting `guidance_type`:
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1. "uncond": unconditional sampling by DPMs.
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The input `model` has the following format:
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``
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model(x, t_input, **model_kwargs) -> noise | x_start | v | score
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``
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2. "classifier": classifier guidance sampling [3] by DPMs and another classifier.
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The input `model` has the following format:
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``
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model(x, t_input, **model_kwargs) -> noise | x_start | v | score
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``
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The input `classifier_fn` has the following format:
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``
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classifier_fn(x, t_input, cond, **classifier_kwargs) -> logits(x, t_input, cond)
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``
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[3] P. Dhariwal and A. Q. Nichol, "Diffusion models beat GANs on image synthesis,"
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in Advances in Neural Information Processing Systems, vol. 34, 2021, pp. 8780-8794.
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3. "classifier-free": classifier-free guidance sampling by conditional DPMs.
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The input `model` has the following format:
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``
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model(x, t_input, cond, **model_kwargs) -> noise | x_start | v | score
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``
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And if cond == `unconditional_condition`, the model output is the unconditional DPM output.
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[4] Ho, Jonathan, and Tim Salimans. "Classifier-free diffusion guidance."
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arXiv preprint arXiv:2207.12598 (2022).
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The `t_input` is the time label of the model, which may be discrete-time labels (i.e. 0 to 999)
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or continuous-time labels (i.e. epsilon to T).
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We wrap the model function to accept only `x` and `t_continuous` as inputs, and outputs the predicted noise:
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``
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def model_fn(x, t_continuous) -> noise:
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t_input = get_model_input_time(t_continuous)
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return noise_pred(model, x, t_input, **model_kwargs)
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``
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where `t_continuous` is the continuous time labels (i.e. epsilon to T). And we use `model_fn` for DPM-Solver.
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===============================================================
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Args:
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model: A diffusion model with the corresponding format described above.
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noise_schedule: A noise schedule object, such as NoiseScheduleVP.
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model_type: A `str`. The parameterization type of the diffusion model.
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"noise" or "x_start" or "v" or "score".
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model_kwargs: A `dict`. A dict for the other inputs of the model function.
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guidance_type: A `str`. The type of the guidance for sampling.
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"uncond" or "classifier" or "classifier-free".
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condition: A pytorch tensor. The condition for the guided sampling.
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Only used for "classifier" or "classifier-free" guidance type.
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unconditional_condition: A pytorch tensor. The condition for the unconditional sampling.
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Only used for "classifier-free" guidance type.
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guidance_scale: A `float`. The scale for the guided sampling.
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classifier_fn: A classifier function. Only used for the classifier guidance.
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classifier_kwargs: A `dict`. A dict for the other inputs of the classifier function.
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Returns:
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A noise prediction model that accepts the noised data and the continuous time as the inputs.
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"""
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def get_model_input_time(t_continuous):
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"""
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Convert the continuous-time `t_continuous` (in [epsilon, T]) to the model input time.
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For discrete-time DPMs, we convert `t_continuous` in [1 / N, 1] to `t_input` in [0, 1000 * (N - 1) / N].
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For continuous-time DPMs, we just use `t_continuous`.
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"""
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if noise_schedule.schedule == 'discrete':
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return (t_continuous - 1. / noise_schedule.total_N) * noise_schedule.total_N
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else:
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return t_continuous
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def noise_pred_fn(x, t_continuous, cond=None):
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t_input = get_model_input_time(t_continuous)
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if cond is None:
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output = model(x, t_input, **model_kwargs)
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else:
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output = model(x, t_input, cond, **model_kwargs)
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if model_type == "noise":
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return output
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elif model_type == "x_start":
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alpha_t, sigma_t = noise_schedule.marginal_alpha(t_continuous), noise_schedule.marginal_std(t_continuous)
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return (x - expand_dims(alpha_t, x.dim()) * output) / expand_dims(sigma_t, x.dim())
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elif model_type == "v":
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alpha_t, sigma_t = noise_schedule.marginal_alpha(t_continuous), noise_schedule.marginal_std(t_continuous)
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return expand_dims(alpha_t, x.dim()) * output + expand_dims(sigma_t, x.dim()) * x
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elif model_type == "score":
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sigma_t = noise_schedule.marginal_std(t_continuous)
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return -expand_dims(sigma_t, x.dim()) * output
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def cond_grad_fn(x, t_input):
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"""
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Compute the gradient of the classifier, i.e. nabla_{x} log p_t(cond | x_t).
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"""
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with torch.enable_grad():
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x_in = x.detach().requires_grad_(True)
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log_prob = classifier_fn(x_in, t_input, condition, **classifier_kwargs)
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return torch.autograd.grad(log_prob.sum(), x_in)[0]
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def model_fn(x, t_continuous):
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"""
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The noise predicition model function that is used for DPM-Solver.
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"""
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if guidance_type == "uncond":
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return noise_pred_fn(x, t_continuous)
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elif guidance_type == "classifier":
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assert classifier_fn is not None
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t_input = get_model_input_time(t_continuous)
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cond_grad = cond_grad_fn(x, t_input)
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sigma_t = noise_schedule.marginal_std(t_continuous)
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noise = noise_pred_fn(x, t_continuous)
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return noise - guidance_scale * expand_dims(sigma_t, x.dim()) * cond_grad
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elif guidance_type == "classifier-free":
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if guidance_scale == 1. or unconditional_condition is None:
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return noise_pred_fn(x, t_continuous, cond=condition)
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else:
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x_in = torch.cat([x] * 2)
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t_in = torch.cat([t_continuous] * 2)
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c_in = torch.cat([unconditional_condition, condition])
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noise_uncond, noise = noise_pred_fn(x_in, t_in, cond=c_in).chunk(2)
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return noise_uncond + guidance_scale * (noise - noise_uncond)
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assert model_type in ["noise", "x_start", "v", "score"]
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assert guidance_type in ["uncond", "classifier", "classifier-free"]
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return model_fn
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class DPM_Solver:
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def __init__(
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self,
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model_fn,
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noise_schedule,
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algorithm_type="dpmsolver++",
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correcting_x0_fn=None,
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correcting_xt_fn=None,
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thresholding_max_val=1.,
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dynamic_thresholding_ratio=0.995,
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):
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"""Construct a DPM-Solver.
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We support both DPM-Solver (`algorithm_type="dpmsolver"`) and DPM-Solver++ (`algorithm_type="dpmsolver++"`).
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We also support the "dynamic thresholding" method in Imagen[1]. For pixel-space diffusion models, you
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can set both `algorithm_type="dpmsolver++"` and `correcting_x0_fn="dynamic_thresholding"` to use the
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dynamic thresholding. The "dynamic thresholding" can greatly improve the sample quality for pixel-space
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DPMs with large guidance scales. Note that the thresholding method is **unsuitable** for latent-space
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DPMs (such as stable-diffusion).
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To support advanced algorithms in image-to-image applications, we also support corrector functions for
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both x0 and xt.
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Args:
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model_fn: A noise prediction model function which accepts the continuous-time input (t in [epsilon, T]):
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``
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def model_fn(x, t_continuous):
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return noise
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``
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The shape of `x` is `(batch_size, **shape)`, and the shape of `t_continuous` is `(batch_size,)`.
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noise_schedule: A noise schedule object, such as NoiseScheduleVP.
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algorithm_type: A `str`. Either "dpmsolver" or "dpmsolver++".
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correcting_x0_fn: A `str` or a function with the following format:
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```
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def correcting_x0_fn(x0, t):
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x0_new = ...
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return x0_new
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```
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This function is to correct the outputs of the data prediction model at each sampling step. e.g.,
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```
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x0_pred = data_pred_model(xt, t)
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if correcting_x0_fn is not None:
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x0_pred = correcting_x0_fn(x0_pred, t)
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xt_1 = update(x0_pred, xt, t)
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```
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If `correcting_x0_fn="dynamic_thresholding"`, we use the dynamic thresholding proposed in Imagen[1].
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correcting_xt_fn: A function with the following format:
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```
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def correcting_xt_fn(xt, t, step):
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x_new = ...
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return x_new
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```
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This function is to correct the intermediate samples xt at each sampling step. e.g.,
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```
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xt = ...
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xt = correcting_xt_fn(xt, t, step)
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```
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thresholding_max_val: A `float`. The max value for thresholding.
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Valid only when use `dpmsolver++` and `correcting_x0_fn="dynamic_thresholding"`.
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dynamic_thresholding_ratio: A `float`. The ratio for dynamic thresholding (see Imagen[1] for details).
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Valid only when use `dpmsolver++` and `correcting_x0_fn="dynamic_thresholding"`.
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[1] Chitwan Saharia, William Chan, Saurabh Saxena, Lala Li, Jay Whang, Emily Denton, Seyed Kamyar Seyed Ghasemipour,
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Burcu Karagol Ayan, S Sara Mahdavi, Rapha Gontijo Lopes, et al. Photorealistic text-to-image diffusion models
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with deep language understanding. arXiv preprint arXiv:2205.11487, 2022b.
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"""
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self.model = lambda x, t: model_fn(x, t.expand((x.shape[0])))
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self.noise_schedule = noise_schedule
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assert algorithm_type in ["dpmsolver", "dpmsolver++"]
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self.algorithm_type = algorithm_type
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if correcting_x0_fn == "dynamic_thresholding":
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self.correcting_x0_fn = self.dynamic_thresholding_fn
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else:
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self.correcting_x0_fn = correcting_x0_fn
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self.correcting_xt_fn = correcting_xt_fn
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self.dynamic_thresholding_ratio = dynamic_thresholding_ratio
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self.thresholding_max_val = thresholding_max_val
|
|
|
|
def dynamic_thresholding_fn(self, x0, t):
|
|
"""
|
|
The dynamic thresholding method.
|
|
"""
|
|
dims = x0.dim()
|
|
p = self.dynamic_thresholding_ratio
|
|
s = torch.quantile(torch.abs(x0).reshape((x0.shape[0], -1)), p, dim=1)
|
|
s = expand_dims(torch.maximum(s, self.thresholding_max_val * torch.ones_like(s).to(s.device)), dims)
|
|
x0 = torch.clamp(x0, -s, s) / s
|
|
return x0
|
|
|
|
def noise_prediction_fn(self, x, t):
|
|
"""
|
|
Return the noise prediction model.
|
|
"""
|
|
return self.model(x, t)
|
|
|
|
def data_prediction_fn(self, x, t):
|
|
"""
|
|
Return the data prediction model (with corrector).
|
|
"""
|
|
noise = self.noise_prediction_fn(x, t)
|
|
alpha_t, sigma_t = self.noise_schedule.marginal_alpha(t), self.noise_schedule.marginal_std(t)
|
|
x0 = (x - sigma_t * noise) / alpha_t
|
|
if self.correcting_x0_fn is not None:
|
|
x0 = self.correcting_x0_fn(x0, t)
|
|
return x0
|
|
|
|
def model_fn(self, x, t):
|
|
"""
|
|
Convert the model to the noise prediction model or the data prediction model.
|
|
"""
|
|
if self.algorithm_type == "dpmsolver++":
|
|
return self.data_prediction_fn(x, t)
|
|
else:
|
|
return self.noise_prediction_fn(x, t)
|
|
|
|
def get_time_steps(self, skip_type, t_T, t_0, N, device):
|
|
"""Compute the intermediate time steps for sampling.
|
|
|
|
Args:
|
|
skip_type: A `str`. The type for the spacing of the time steps. We support three types:
|
|
- 'logSNR': uniform logSNR for the time steps.
|
|
- 'time_uniform': uniform time for the time steps. (**Recommended for high-resolutional data**.)
|
|
- 'time_quadratic': quadratic time for the time steps. (Used in DDIM for low-resolutional data.)
|
|
t_T: A `float`. The starting time of the sampling (default is T).
|
|
t_0: A `float`. The ending time of the sampling (default is epsilon).
|
|
N: A `int`. The total number of the spacing of the time steps.
|
|
device: A torch device.
|
|
Returns:
|
|
A pytorch tensor of the time steps, with the shape (N + 1,).
|
|
"""
|
|
if skip_type == 'logSNR':
|
|
lambda_T = self.noise_schedule.marginal_lambda(torch.tensor(t_T).to(device))
|
|
lambda_0 = self.noise_schedule.marginal_lambda(torch.tensor(t_0).to(device))
|
|
logSNR_steps = torch.linspace(lambda_T.cpu().item(), lambda_0.cpu().item(), N + 1).to(device)
|
|
return self.noise_schedule.inverse_lambda(logSNR_steps)
|
|
elif skip_type == 'time_uniform':
|
|
return torch.linspace(t_T, t_0, N + 1).to(device)
|
|
elif skip_type == 'time_quadratic':
|
|
t_order = 2
|
|
t = torch.linspace(t_T**(1. / t_order), t_0**(1. / t_order), N + 1).pow(t_order).to(device)
|
|
return t
|
|
else:
|
|
raise ValueError("Unsupported skip_type {}, need to be 'logSNR' or 'time_uniform' or 'time_quadratic'".format(skip_type))
|
|
|
|
def get_orders_and_timesteps_for_singlestep_solver(self, steps, order, skip_type, t_T, t_0, device):
|
|
"""
|
|
Get the order of each step for sampling by the singlestep DPM-Solver.
|
|
|
|
We combine both DPM-Solver-1,2,3 to use all the function evaluations, which is named as "DPM-Solver-fast".
|
|
Given a fixed number of function evaluations by `steps`, the sampling procedure by DPM-Solver-fast is:
|
|
- If order == 1:
|
|
We take `steps` of DPM-Solver-1 (i.e. DDIM).
|
|
- If order == 2:
|
|
- Denote K = (steps // 2). We take K or (K + 1) intermediate time steps for sampling.
|
|
- If steps % 2 == 0, we use K steps of DPM-Solver-2.
|
|
- If steps % 2 == 1, we use K steps of DPM-Solver-2 and 1 step of DPM-Solver-1.
|
|
- If order == 3:
|
|
- Denote K = (steps // 3 + 1). We take K intermediate time steps for sampling.
|
|
- If steps % 3 == 0, we use (K - 2) steps of DPM-Solver-3, and 1 step of DPM-Solver-2 and 1 step of DPM-Solver-1.
|
|
- If steps % 3 == 1, we use (K - 1) steps of DPM-Solver-3 and 1 step of DPM-Solver-1.
|
|
- If steps % 3 == 2, we use (K - 1) steps of DPM-Solver-3 and 1 step of DPM-Solver-2.
|
|
|
|
============================================
|
|
Args:
|
|
order: A `int`. The max order for the solver (2 or 3).
|
|
steps: A `int`. The total number of function evaluations (NFE).
|
|
skip_type: A `str`. The type for the spacing of the time steps. We support three types:
|
|
- 'logSNR': uniform logSNR for the time steps.
|
|
- 'time_uniform': uniform time for the time steps. (**Recommended for high-resolutional data**.)
|
|
- 'time_quadratic': quadratic time for the time steps. (Used in DDIM for low-resolutional data.)
|
|
t_T: A `float`. The starting time of the sampling (default is T).
|
|
t_0: A `float`. The ending time of the sampling (default is epsilon).
|
|
device: A torch device.
|
|
Returns:
|
|
orders: A list of the solver order of each step.
|
|
"""
|
|
if order == 3:
|
|
K = steps // 3 + 1
|
|
if steps % 3 == 0:
|
|
orders = [3,] * (K - 2) + [2, 1]
|
|
elif steps % 3 == 1:
|
|
orders = [3,] * (K - 1) + [1]
|
|
else:
|
|
orders = [3,] * (K - 1) + [2]
|
|
elif order == 2:
|
|
if steps % 2 == 0:
|
|
K = steps // 2
|
|
orders = [2,] * K
|
|
else:
|
|
K = steps // 2 + 1
|
|
orders = [2,] * (K - 1) + [1]
|
|
elif order == 1:
|
|
K = 1
|
|
orders = [1,] * steps
|
|
else:
|
|
raise ValueError("'order' must be '1' or '2' or '3'.")
|
|
if skip_type == 'logSNR':
|
|
# To reproduce the results in DPM-Solver paper
|
|
timesteps_outer = self.get_time_steps(skip_type, t_T, t_0, K, device)
|
|
else:
|
|
timesteps_outer = self.get_time_steps(skip_type, t_T, t_0, steps, device)[torch.cumsum(torch.tensor([0,] + orders), 0).to(device)]
|
|
return timesteps_outer, orders
|
|
|
|
def denoise_to_zero_fn(self, x, s):
|
|
"""
|
|
Denoise at the final step, which is equivalent to solve the ODE from lambda_s to infty by first-order discretization.
|
|
"""
|
|
return self.data_prediction_fn(x, s)
|
|
|
|
def dpm_solver_first_update(self, x, s, t, model_s=None, return_intermediate=False):
|
|
"""
|
|
DPM-Solver-1 (equivalent to DDIM) from time `s` to time `t`.
|
|
|
|
Args:
|
|
x: A pytorch tensor. The initial value at time `s`.
|
|
s: A pytorch tensor. The starting time, with the shape (1,).
|
|
t: A pytorch tensor. The ending time, with the shape (1,).
|
|
model_s: A pytorch tensor. The model function evaluated at time `s`.
|
|
If `model_s` is None, we evaluate the model by `x` and `s`; otherwise we directly use it.
|
|
return_intermediate: A `bool`. If true, also return the model value at time `s`.
|
|
Returns:
|
|
x_t: A pytorch tensor. The approximated solution at time `t`.
|
|
"""
|
|
ns = self.noise_schedule
|
|
lambda_s, lambda_t = ns.marginal_lambda(s), ns.marginal_lambda(t)
|
|
h = lambda_t - lambda_s
|
|
log_alpha_s, log_alpha_t = ns.marginal_log_mean_coeff(s), ns.marginal_log_mean_coeff(t)
|
|
sigma_s, sigma_t = ns.marginal_std(s), ns.marginal_std(t)
|
|
alpha_t = torch.exp(log_alpha_t)
|
|
|
|
if self.algorithm_type == "dpmsolver++":
|
|
phi_1 = torch.expm1(-h)
|
|
if model_s is None:
|
|
model_s = self.model_fn(x, s)
|
|
x_t = (
|
|
sigma_t / sigma_s * x
|
|
- alpha_t * phi_1 * model_s
|
|
)
|
|
if return_intermediate:
|
|
return x_t, {'model_s': model_s}
|
|
else:
|
|
return x_t
|
|
else:
|
|
phi_1 = torch.expm1(h)
|
|
if model_s is None:
|
|
model_s = self.model_fn(x, s)
|
|
x_t = (
|
|
torch.exp(log_alpha_t - log_alpha_s) * x
|
|
- (sigma_t * phi_1) * model_s
|
|
)
|
|
if return_intermediate:
|
|
return x_t, {'model_s': model_s}
|
|
else:
|
|
return x_t
|
|
|
|
def singlestep_dpm_solver_second_update(self, x, s, t, r1=0.5, model_s=None, return_intermediate=False, solver_type='dpmsolver'):
|
|
"""
|
|
Singlestep solver DPM-Solver-2 from time `s` to time `t`.
|
|
|
|
Args:
|
|
x: A pytorch tensor. The initial value at time `s`.
|
|
s: A pytorch tensor. The starting time, with the shape (1,).
|
|
t: A pytorch tensor. The ending time, with the shape (1,).
|
|
r1: A `float`. The hyperparameter of the second-order solver.
|
|
model_s: A pytorch tensor. The model function evaluated at time `s`.
|
|
If `model_s` is None, we evaluate the model by `x` and `s`; otherwise we directly use it.
|
|
return_intermediate: A `bool`. If true, also return the model value at time `s` and `s1` (the intermediate time).
|
|
solver_type: either 'dpmsolver' or 'taylor'. The type for the high-order solvers.
|
|
The type slightly impacts the performance. We recommend to use 'dpmsolver' type.
|
|
Returns:
|
|
x_t: A pytorch tensor. The approximated solution at time `t`.
|
|
"""
|
|
if solver_type not in ['dpmsolver', 'taylor']:
|
|
raise ValueError("'solver_type' must be either 'dpmsolver' or 'taylor', got {}".format(solver_type))
|
|
if r1 is None:
|
|
r1 = 0.5
|
|
ns = self.noise_schedule
|
|
lambda_s, lambda_t = ns.marginal_lambda(s), ns.marginal_lambda(t)
|
|
h = lambda_t - lambda_s
|
|
lambda_s1 = lambda_s + r1 * h
|
|
s1 = ns.inverse_lambda(lambda_s1)
|
|
log_alpha_s, log_alpha_s1, log_alpha_t = ns.marginal_log_mean_coeff(s), ns.marginal_log_mean_coeff(s1), ns.marginal_log_mean_coeff(t)
|
|
sigma_s, sigma_s1, sigma_t = ns.marginal_std(s), ns.marginal_std(s1), ns.marginal_std(t)
|
|
alpha_s1, alpha_t = torch.exp(log_alpha_s1), torch.exp(log_alpha_t)
|
|
|
|
if self.algorithm_type == "dpmsolver++":
|
|
phi_11 = torch.expm1(-r1 * h)
|
|
phi_1 = torch.expm1(-h)
|
|
|
|
if model_s is None:
|
|
model_s = self.model_fn(x, s)
|
|
x_s1 = (
|
|
(sigma_s1 / sigma_s) * x
|
|
- (alpha_s1 * phi_11) * model_s
|
|
)
|
|
model_s1 = self.model_fn(x_s1, s1)
|
|
if solver_type == 'dpmsolver':
|
|
x_t = (
|
|
(sigma_t / sigma_s) * x
|
|
- (alpha_t * phi_1) * model_s
|
|
- (0.5 / r1) * (alpha_t * phi_1) * (model_s1 - model_s)
|
|
)
|
|
elif solver_type == 'taylor':
|
|
x_t = (
|
|
(sigma_t / sigma_s) * x
|
|
- (alpha_t * phi_1) * model_s
|
|
+ (1. / r1) * (alpha_t * (phi_1 / h + 1.)) * (model_s1 - model_s)
|
|
)
|
|
else:
|
|
phi_11 = torch.expm1(r1 * h)
|
|
phi_1 = torch.expm1(h)
|
|
|
|
if model_s is None:
|
|
model_s = self.model_fn(x, s)
|
|
x_s1 = (
|
|
torch.exp(log_alpha_s1 - log_alpha_s) * x
|
|
- (sigma_s1 * phi_11) * model_s
|
|
)
|
|
model_s1 = self.model_fn(x_s1, s1)
|
|
if solver_type == 'dpmsolver':
|
|
x_t = (
|
|
torch.exp(log_alpha_t - log_alpha_s) * x
|
|
- (sigma_t * phi_1) * model_s
|
|
- (0.5 / r1) * (sigma_t * phi_1) * (model_s1 - model_s)
|
|
)
|
|
elif solver_type == 'taylor':
|
|
x_t = (
|
|
torch.exp(log_alpha_t - log_alpha_s) * x
|
|
- (sigma_t * phi_1) * model_s
|
|
- (1. / r1) * (sigma_t * (phi_1 / h - 1.)) * (model_s1 - model_s)
|
|
)
|
|
if return_intermediate:
|
|
return x_t, {'model_s': model_s, 'model_s1': model_s1}
|
|
else:
|
|
return x_t
|
|
|
|
def singlestep_dpm_solver_third_update(self, x, s, t, r1=1./3., r2=2./3., model_s=None, model_s1=None, return_intermediate=False, solver_type='dpmsolver'):
|
|
"""
|
|
Singlestep solver DPM-Solver-3 from time `s` to time `t`.
|
|
|
|
Args:
|
|
x: A pytorch tensor. The initial value at time `s`.
|
|
s: A pytorch tensor. The starting time, with the shape (1,).
|
|
t: A pytorch tensor. The ending time, with the shape (1,).
|
|
r1: A `float`. The hyperparameter of the third-order solver.
|
|
r2: A `float`. The hyperparameter of the third-order solver.
|
|
model_s: A pytorch tensor. The model function evaluated at time `s`.
|
|
If `model_s` is None, we evaluate the model by `x` and `s`; otherwise we directly use it.
|
|
model_s1: A pytorch tensor. The model function evaluated at time `s1` (the intermediate time given by `r1`).
|
|
If `model_s1` is None, we evaluate the model at `s1`; otherwise we directly use it.
|
|
return_intermediate: A `bool`. If true, also return the model value at time `s`, `s1` and `s2` (the intermediate times).
|
|
solver_type: either 'dpmsolver' or 'taylor'. The type for the high-order solvers.
|
|
The type slightly impacts the performance. We recommend to use 'dpmsolver' type.
|
|
Returns:
|
|
x_t: A pytorch tensor. The approximated solution at time `t`.
|
|
"""
|
|
if solver_type not in ['dpmsolver', 'taylor']:
|
|
raise ValueError("'solver_type' must be either 'dpmsolver' or 'taylor', got {}".format(solver_type))
|
|
if r1 is None:
|
|
r1 = 1. / 3.
|
|
if r2 is None:
|
|
r2 = 2. / 3.
|
|
ns = self.noise_schedule
|
|
lambda_s, lambda_t = ns.marginal_lambda(s), ns.marginal_lambda(t)
|
|
h = lambda_t - lambda_s
|
|
lambda_s1 = lambda_s + r1 * h
|
|
lambda_s2 = lambda_s + r2 * h
|
|
s1 = ns.inverse_lambda(lambda_s1)
|
|
s2 = ns.inverse_lambda(lambda_s2)
|
|
log_alpha_s, log_alpha_s1, log_alpha_s2, log_alpha_t = ns.marginal_log_mean_coeff(s), ns.marginal_log_mean_coeff(s1), ns.marginal_log_mean_coeff(s2), ns.marginal_log_mean_coeff(t)
|
|
sigma_s, sigma_s1, sigma_s2, sigma_t = ns.marginal_std(s), ns.marginal_std(s1), ns.marginal_std(s2), ns.marginal_std(t)
|
|
alpha_s1, alpha_s2, alpha_t = torch.exp(log_alpha_s1), torch.exp(log_alpha_s2), torch.exp(log_alpha_t)
|
|
|
|
if self.algorithm_type == "dpmsolver++":
|
|
phi_11 = torch.expm1(-r1 * h)
|
|
phi_12 = torch.expm1(-r2 * h)
|
|
phi_1 = torch.expm1(-h)
|
|
phi_22 = torch.expm1(-r2 * h) / (r2 * h) + 1.
|
|
phi_2 = phi_1 / h + 1.
|
|
phi_3 = phi_2 / h - 0.5
|
|
|
|
if model_s is None:
|
|
model_s = self.model_fn(x, s)
|
|
if model_s1 is None:
|
|
x_s1 = (
|
|
(sigma_s1 / sigma_s) * x
|
|
- (alpha_s1 * phi_11) * model_s
|
|
)
|
|
model_s1 = self.model_fn(x_s1, s1)
|
|
x_s2 = (
|
|
(sigma_s2 / sigma_s) * x
|
|
- (alpha_s2 * phi_12) * model_s
|
|
+ r2 / r1 * (alpha_s2 * phi_22) * (model_s1 - model_s)
|
|
)
|
|
model_s2 = self.model_fn(x_s2, s2)
|
|
if solver_type == 'dpmsolver':
|
|
x_t = (
|
|
(sigma_t / sigma_s) * x
|
|
- (alpha_t * phi_1) * model_s
|
|
+ (1. / r2) * (alpha_t * phi_2) * (model_s2 - model_s)
|
|
)
|
|
elif solver_type == 'taylor':
|
|
D1_0 = (1. / r1) * (model_s1 - model_s)
|
|
D1_1 = (1. / r2) * (model_s2 - model_s)
|
|
D1 = (r2 * D1_0 - r1 * D1_1) / (r2 - r1)
|
|
D2 = 2. * (D1_1 - D1_0) / (r2 - r1)
|
|
x_t = (
|
|
(sigma_t / sigma_s) * x
|
|
- (alpha_t * phi_1) * model_s
|
|
+ (alpha_t * phi_2) * D1
|
|
- (alpha_t * phi_3) * D2
|
|
)
|
|
else:
|
|
phi_11 = torch.expm1(r1 * h)
|
|
phi_12 = torch.expm1(r2 * h)
|
|
phi_1 = torch.expm1(h)
|
|
phi_22 = torch.expm1(r2 * h) / (r2 * h) - 1.
|
|
phi_2 = phi_1 / h - 1.
|
|
phi_3 = phi_2 / h - 0.5
|
|
|
|
if model_s is None:
|
|
model_s = self.model_fn(x, s)
|
|
if model_s1 is None:
|
|
x_s1 = (
|
|
(torch.exp(log_alpha_s1 - log_alpha_s)) * x
|
|
- (sigma_s1 * phi_11) * model_s
|
|
)
|
|
model_s1 = self.model_fn(x_s1, s1)
|
|
x_s2 = (
|
|
(torch.exp(log_alpha_s2 - log_alpha_s)) * x
|
|
- (sigma_s2 * phi_12) * model_s
|
|
- r2 / r1 * (sigma_s2 * phi_22) * (model_s1 - model_s)
|
|
)
|
|
model_s2 = self.model_fn(x_s2, s2)
|
|
if solver_type == 'dpmsolver':
|
|
x_t = (
|
|
(torch.exp(log_alpha_t - log_alpha_s)) * x
|
|
- (sigma_t * phi_1) * model_s
|
|
- (1. / r2) * (sigma_t * phi_2) * (model_s2 - model_s)
|
|
)
|
|
elif solver_type == 'taylor':
|
|
D1_0 = (1. / r1) * (model_s1 - model_s)
|
|
D1_1 = (1. / r2) * (model_s2 - model_s)
|
|
D1 = (r2 * D1_0 - r1 * D1_1) / (r2 - r1)
|
|
D2 = 2. * (D1_1 - D1_0) / (r2 - r1)
|
|
x_t = (
|
|
(torch.exp(log_alpha_t - log_alpha_s)) * x
|
|
- (sigma_t * phi_1) * model_s
|
|
- (sigma_t * phi_2) * D1
|
|
- (sigma_t * phi_3) * D2
|
|
)
|
|
|
|
if return_intermediate:
|
|
return x_t, {'model_s': model_s, 'model_s1': model_s1, 'model_s2': model_s2}
|
|
else:
|
|
return x_t
|
|
|
|
def multistep_dpm_solver_second_update(self, x, model_prev_list, t_prev_list, t, solver_type="dpmsolver"):
|
|
"""
|
|
Multistep solver DPM-Solver-2 from time `t_prev_list[-1]` to time `t`.
|
|
|
|
Args:
|
|
x: A pytorch tensor. The initial value at time `s`.
|
|
model_prev_list: A list of pytorch tensor. The previous computed model values.
|
|
t_prev_list: A list of pytorch tensor. The previous times, each time has the shape (1,)
|
|
t: A pytorch tensor. The ending time, with the shape (1,).
|
|
solver_type: either 'dpmsolver' or 'taylor'. The type for the high-order solvers.
|
|
The type slightly impacts the performance. We recommend to use 'dpmsolver' type.
|
|
Returns:
|
|
x_t: A pytorch tensor. The approximated solution at time `t`.
|
|
"""
|
|
if solver_type not in ['dpmsolver', 'taylor']:
|
|
raise ValueError("'solver_type' must be either 'dpmsolver' or 'taylor', got {}".format(solver_type))
|
|
ns = self.noise_schedule
|
|
model_prev_1, model_prev_0 = model_prev_list[-2], model_prev_list[-1]
|
|
t_prev_1, t_prev_0 = t_prev_list[-2], t_prev_list[-1]
|
|
lambda_prev_1, lambda_prev_0, lambda_t = ns.marginal_lambda(t_prev_1), ns.marginal_lambda(t_prev_0), ns.marginal_lambda(t)
|
|
log_alpha_prev_0, log_alpha_t = ns.marginal_log_mean_coeff(t_prev_0), ns.marginal_log_mean_coeff(t)
|
|
sigma_prev_0, sigma_t = ns.marginal_std(t_prev_0), ns.marginal_std(t)
|
|
alpha_t = torch.exp(log_alpha_t)
|
|
|
|
h_0 = lambda_prev_0 - lambda_prev_1
|
|
h = lambda_t - lambda_prev_0
|
|
r0 = h_0 / h
|
|
D1_0 = (1. / r0) * (model_prev_0 - model_prev_1)
|
|
if self.algorithm_type == "dpmsolver++":
|
|
phi_1 = torch.expm1(-h)
|
|
if solver_type == 'dpmsolver':
|
|
x_t = (
|
|
(sigma_t / sigma_prev_0) * x
|
|
- (alpha_t * phi_1) * model_prev_0
|
|
- 0.5 * (alpha_t * phi_1) * D1_0
|
|
)
|
|
elif solver_type == 'taylor':
|
|
x_t = (
|
|
(sigma_t / sigma_prev_0) * x
|
|
- (alpha_t * phi_1) * model_prev_0
|
|
+ (alpha_t * (phi_1 / h + 1.)) * D1_0
|
|
)
|
|
else:
|
|
phi_1 = torch.expm1(h)
|
|
if solver_type == 'dpmsolver':
|
|
x_t = (
|
|
(torch.exp(log_alpha_t - log_alpha_prev_0)) * x
|
|
- (sigma_t * phi_1) * model_prev_0
|
|
- 0.5 * (sigma_t * phi_1) * D1_0
|
|
)
|
|
elif solver_type == 'taylor':
|
|
x_t = (
|
|
(torch.exp(log_alpha_t - log_alpha_prev_0)) * x
|
|
- (sigma_t * phi_1) * model_prev_0
|
|
- (sigma_t * (phi_1 / h - 1.)) * D1_0
|
|
)
|
|
return x_t
|
|
|
|
def multistep_dpm_solver_third_update(self, x, model_prev_list, t_prev_list, t, solver_type='dpmsolver'):
|
|
"""
|
|
Multistep solver DPM-Solver-3 from time `t_prev_list[-1]` to time `t`.
|
|
|
|
Args:
|
|
x: A pytorch tensor. The initial value at time `s`.
|
|
model_prev_list: A list of pytorch tensor. The previous computed model values.
|
|
t_prev_list: A list of pytorch tensor. The previous times, each time has the shape (1,)
|
|
t: A pytorch tensor. The ending time, with the shape (1,).
|
|
solver_type: either 'dpmsolver' or 'taylor'. The type for the high-order solvers.
|
|
The type slightly impacts the performance. We recommend to use 'dpmsolver' type.
|
|
Returns:
|
|
x_t: A pytorch tensor. The approximated solution at time `t`.
|
|
"""
|
|
ns = self.noise_schedule
|
|
model_prev_2, model_prev_1, model_prev_0 = model_prev_list
|
|
t_prev_2, t_prev_1, t_prev_0 = t_prev_list
|
|
lambda_prev_2, lambda_prev_1, lambda_prev_0, lambda_t = ns.marginal_lambda(t_prev_2), ns.marginal_lambda(t_prev_1), ns.marginal_lambda(t_prev_0), ns.marginal_lambda(t)
|
|
log_alpha_prev_0, log_alpha_t = ns.marginal_log_mean_coeff(t_prev_0), ns.marginal_log_mean_coeff(t)
|
|
sigma_prev_0, sigma_t = ns.marginal_std(t_prev_0), ns.marginal_std(t)
|
|
alpha_t = torch.exp(log_alpha_t)
|
|
|
|
h_1 = lambda_prev_1 - lambda_prev_2
|
|
h_0 = lambda_prev_0 - lambda_prev_1
|
|
h = lambda_t - lambda_prev_0
|
|
r0, r1 = h_0 / h, h_1 / h
|
|
D1_0 = (1. / r0) * (model_prev_0 - model_prev_1)
|
|
D1_1 = (1. / r1) * (model_prev_1 - model_prev_2)
|
|
D1 = D1_0 + (r0 / (r0 + r1)) * (D1_0 - D1_1)
|
|
D2 = (1. / (r0 + r1)) * (D1_0 - D1_1)
|
|
if self.algorithm_type == "dpmsolver++":
|
|
phi_1 = torch.expm1(-h)
|
|
phi_2 = phi_1 / h + 1.
|
|
phi_3 = phi_2 / h - 0.5
|
|
x_t = (
|
|
(sigma_t / sigma_prev_0) * x
|
|
- (alpha_t * phi_1) * model_prev_0
|
|
+ (alpha_t * phi_2) * D1
|
|
- (alpha_t * phi_3) * D2
|
|
)
|
|
else:
|
|
phi_1 = torch.expm1(h)
|
|
phi_2 = phi_1 / h - 1.
|
|
phi_3 = phi_2 / h - 0.5
|
|
x_t = (
|
|
(torch.exp(log_alpha_t - log_alpha_prev_0)) * x
|
|
- (sigma_t * phi_1) * model_prev_0
|
|
- (sigma_t * phi_2) * D1
|
|
- (sigma_t * phi_3) * D2
|
|
)
|
|
return x_t
|
|
|
|
def singlestep_dpm_solver_update(self, x, s, t, order, return_intermediate=False, solver_type='dpmsolver', r1=None, r2=None):
|
|
"""
|
|
Singlestep DPM-Solver with the order `order` from time `s` to time `t`.
|
|
|
|
Args:
|
|
x: A pytorch tensor. The initial value at time `s`.
|
|
s: A pytorch tensor. The starting time, with the shape (1,).
|
|
t: A pytorch tensor. The ending time, with the shape (1,).
|
|
order: A `int`. The order of DPM-Solver. We only support order == 1 or 2 or 3.
|
|
return_intermediate: A `bool`. If true, also return the model value at time `s`, `s1` and `s2` (the intermediate times).
|
|
solver_type: either 'dpmsolver' or 'taylor'. The type for the high-order solvers.
|
|
The type slightly impacts the performance. We recommend to use 'dpmsolver' type.
|
|
r1: A `float`. The hyperparameter of the second-order or third-order solver.
|
|
r2: A `float`. The hyperparameter of the third-order solver.
|
|
Returns:
|
|
x_t: A pytorch tensor. The approximated solution at time `t`.
|
|
"""
|
|
if order == 1:
|
|
return self.dpm_solver_first_update(x, s, t, return_intermediate=return_intermediate)
|
|
elif order == 2:
|
|
return self.singlestep_dpm_solver_second_update(x, s, t, return_intermediate=return_intermediate, solver_type=solver_type, r1=r1)
|
|
elif order == 3:
|
|
return self.singlestep_dpm_solver_third_update(x, s, t, return_intermediate=return_intermediate, solver_type=solver_type, r1=r1, r2=r2)
|
|
else:
|
|
raise ValueError("Solver order must be 1 or 2 or 3, got {}".format(order))
|
|
|
|
def multistep_dpm_solver_update(self, x, model_prev_list, t_prev_list, t, order, solver_type='dpmsolver'):
|
|
"""
|
|
Multistep DPM-Solver with the order `order` from time `t_prev_list[-1]` to time `t`.
|
|
|
|
Args:
|
|
x: A pytorch tensor. The initial value at time `s`.
|
|
model_prev_list: A list of pytorch tensor. The previous computed model values.
|
|
t_prev_list: A list of pytorch tensor. The previous times, each time has the shape (1,)
|
|
t: A pytorch tensor. The ending time, with the shape (1,).
|
|
order: A `int`. The order of DPM-Solver. We only support order == 1 or 2 or 3.
|
|
solver_type: either 'dpmsolver' or 'taylor'. The type for the high-order solvers.
|
|
The type slightly impacts the performance. We recommend to use 'dpmsolver' type.
|
|
Returns:
|
|
x_t: A pytorch tensor. The approximated solution at time `t`.
|
|
"""
|
|
if order == 1:
|
|
return self.dpm_solver_first_update(x, t_prev_list[-1], t, model_s=model_prev_list[-1])
|
|
elif order == 2:
|
|
return self.multistep_dpm_solver_second_update(x, model_prev_list, t_prev_list, t, solver_type=solver_type)
|
|
elif order == 3:
|
|
return self.multistep_dpm_solver_third_update(x, model_prev_list, t_prev_list, t, solver_type=solver_type)
|
|
else:
|
|
raise ValueError("Solver order must be 1 or 2 or 3, got {}".format(order))
|
|
|
|
def dpm_solver_adaptive(self, x, order, t_T, t_0, h_init=0.05, atol=0.0078, rtol=0.05, theta=0.9, t_err=1e-5, solver_type='dpmsolver'):
|
|
"""
|
|
The adaptive step size solver based on singlestep DPM-Solver.
|
|
|
|
Args:
|
|
x: A pytorch tensor. The initial value at time `t_T`.
|
|
order: A `int`. The (higher) order of the solver. We only support order == 2 or 3.
|
|
t_T: A `float`. The starting time of the sampling (default is T).
|
|
t_0: A `float`. The ending time of the sampling (default is epsilon).
|
|
h_init: A `float`. The initial step size (for logSNR).
|
|
atol: A `float`. The absolute tolerance of the solver. For image data, the default setting is 0.0078, followed [1].
|
|
rtol: A `float`. The relative tolerance of the solver. The default setting is 0.05.
|
|
theta: A `float`. The safety hyperparameter for adapting the step size. The default setting is 0.9, followed [1].
|
|
t_err: A `float`. The tolerance for the time. We solve the diffusion ODE until the absolute error between the
|
|
current time and `t_0` is less than `t_err`. The default setting is 1e-5.
|
|
solver_type: either 'dpmsolver' or 'taylor'. The type for the high-order solvers.
|
|
The type slightly impacts the performance. We recommend to use 'dpmsolver' type.
|
|
Returns:
|
|
x_0: A pytorch tensor. The approximated solution at time `t_0`.
|
|
|
|
[1] A. Jolicoeur-Martineau, K. Li, R. Piché-Taillefer, T. Kachman, and I. Mitliagkas, "Gotta go fast when generating data with score-based models," arXiv preprint arXiv:2105.14080, 2021.
|
|
"""
|
|
ns = self.noise_schedule
|
|
s = t_T * torch.ones((1,)).to(x)
|
|
lambda_s = ns.marginal_lambda(s)
|
|
lambda_0 = ns.marginal_lambda(t_0 * torch.ones_like(s).to(x))
|
|
h = h_init * torch.ones_like(s).to(x)
|
|
x_prev = x
|
|
nfe = 0
|
|
if order == 2:
|
|
r1 = 0.5
|
|
def lower_update(x, s, t):
|
|
return self.dpm_solver_first_update(x, s, t, return_intermediate=True)
|
|
def higher_update(x, s, t, **kwargs):
|
|
return self.singlestep_dpm_solver_second_update(x, s, t, r1=r1, solver_type=solver_type, **kwargs)
|
|
elif order == 3:
|
|
r1, r2 = 1. / 3., 2. / 3.
|
|
def lower_update(x, s, t):
|
|
return self.singlestep_dpm_solver_second_update(x, s, t, r1=r1, return_intermediate=True, solver_type=solver_type)
|
|
def higher_update(x, s, t, **kwargs):
|
|
return self.singlestep_dpm_solver_third_update(x, s, t, r1=r1, r2=r2, solver_type=solver_type, **kwargs)
|
|
else:
|
|
raise ValueError("For adaptive step size solver, order must be 2 or 3, got {}".format(order))
|
|
while torch.abs((s - t_0)).mean() > t_err:
|
|
t = ns.inverse_lambda(lambda_s + h)
|
|
x_lower, lower_noise_kwargs = lower_update(x, s, t)
|
|
x_higher = higher_update(x, s, t, **lower_noise_kwargs)
|
|
delta = torch.max(torch.ones_like(x).to(x) * atol, rtol * torch.max(torch.abs(x_lower), torch.abs(x_prev)))
|
|
def norm_fn(v):
|
|
return torch.sqrt(torch.square(v.reshape((v.shape[0], -1))).mean(dim=-1, keepdim=True))
|
|
E = norm_fn((x_higher - x_lower) / delta).max()
|
|
if torch.all(E <= 1.):
|
|
x = x_higher
|
|
s = t
|
|
x_prev = x_lower
|
|
lambda_s = ns.marginal_lambda(s)
|
|
h = torch.min(theta * h * torch.float_power(E, -1. / order).float(), lambda_0 - lambda_s)
|
|
nfe += order
|
|
print('adaptive solver nfe', nfe)
|
|
return x
|
|
|
|
def add_noise(self, x, t, noise=None):
|
|
"""
|
|
Compute the noised input xt = alpha_t * x + sigma_t * noise.
|
|
|
|
Args:
|
|
x: A `torch.Tensor` with shape `(batch_size, *shape)`.
|
|
t: A `torch.Tensor` with shape `(t_size,)`.
|
|
Returns:
|
|
xt with shape `(t_size, batch_size, *shape)`.
|
|
"""
|
|
alpha_t, sigma_t = self.noise_schedule.marginal_alpha(t), self.noise_schedule.marginal_std(t)
|
|
if noise is None:
|
|
noise = torch.randn((t.shape[0], *x.shape), device=x.device)
|
|
x = x.reshape((-1, *x.shape))
|
|
xt = expand_dims(alpha_t, x.dim()) * x + expand_dims(sigma_t, x.dim()) * noise
|
|
if t.shape[0] == 1:
|
|
return xt.squeeze(0)
|
|
else:
|
|
return xt
|
|
|
|
def inverse(self, x, steps=20, t_start=None, t_end=None, order=2, skip_type='time_uniform',
|
|
method='multistep', lower_order_final=True, denoise_to_zero=False, solver_type='dpmsolver',
|
|
atol=0.0078, rtol=0.05, return_intermediate=False,
|
|
):
|
|
"""
|
|
Inverse the sample `x` from time `t_start` to `t_end` by DPM-Solver.
|
|
For discrete-time DPMs, we use `t_start=1/N`, where `N` is the total time steps during training.
|
|
"""
|
|
t_0 = 1. / self.noise_schedule.total_N if t_start is None else t_start
|
|
t_T = self.noise_schedule.T if t_end is None else t_end
|
|
assert t_0 > 0 and t_T > 0, "Time range needs to be greater than 0. For discrete-time DPMs, it needs to be in [1 / N, 1], where N is the length of betas array"
|
|
return self.sample(x, steps=steps, t_start=t_0, t_end=t_T, order=order, skip_type=skip_type,
|
|
method=method, lower_order_final=lower_order_final, denoise_to_zero=denoise_to_zero, solver_type=solver_type,
|
|
atol=atol, rtol=rtol, return_intermediate=return_intermediate)
|
|
|
|
def sample(self, x, steps=20, t_start=None, t_end=None, order=2, skip_type='time_uniform',
|
|
method='multistep', lower_order_final=True, denoise_to_zero=False, solver_type='dpmsolver',
|
|
atol=0.0078, rtol=0.05, return_intermediate=False,
|
|
):
|
|
"""
|
|
Compute the sample at time `t_end` by DPM-Solver, given the initial `x` at time `t_start`.
|
|
|
|
=====================================================
|
|
|
|
We support the following algorithms for both noise prediction model and data prediction model:
|
|
- 'singlestep':
|
|
Singlestep DPM-Solver (i.e. "DPM-Solver-fast" in the paper), which combines different orders of singlestep DPM-Solver.
|
|
We combine all the singlestep solvers with order <= `order` to use up all the function evaluations (steps).
|
|
The total number of function evaluations (NFE) == `steps`.
|
|
Given a fixed NFE == `steps`, the sampling procedure is:
|
|
- If `order` == 1:
|
|
- Denote K = steps. We use K steps of DPM-Solver-1 (i.e. DDIM).
|
|
- If `order` == 2:
|
|
- Denote K = (steps // 2) + (steps % 2). We take K intermediate time steps for sampling.
|
|
- If steps % 2 == 0, we use K steps of singlestep DPM-Solver-2.
|
|
- If steps % 2 == 1, we use (K - 1) steps of singlestep DPM-Solver-2 and 1 step of DPM-Solver-1.
|
|
- If `order` == 3:
|
|
- Denote K = (steps // 3 + 1). We take K intermediate time steps for sampling.
|
|
- If steps % 3 == 0, we use (K - 2) steps of singlestep DPM-Solver-3, and 1 step of singlestep DPM-Solver-2 and 1 step of DPM-Solver-1.
|
|
- If steps % 3 == 1, we use (K - 1) steps of singlestep DPM-Solver-3 and 1 step of DPM-Solver-1.
|
|
- If steps % 3 == 2, we use (K - 1) steps of singlestep DPM-Solver-3 and 1 step of singlestep DPM-Solver-2.
|
|
- 'multistep':
|
|
Multistep DPM-Solver with the order of `order`. The total number of function evaluations (NFE) == `steps`.
|
|
We initialize the first `order` values by lower order multistep solvers.
|
|
Given a fixed NFE == `steps`, the sampling procedure is:
|
|
Denote K = steps.
|
|
- If `order` == 1:
|
|
- We use K steps of DPM-Solver-1 (i.e. DDIM).
|
|
- If `order` == 2:
|
|
- We firstly use 1 step of DPM-Solver-1, then use (K - 1) step of multistep DPM-Solver-2.
|
|
- If `order` == 3:
|
|
- We firstly use 1 step of DPM-Solver-1, then 1 step of multistep DPM-Solver-2, then (K - 2) step of multistep DPM-Solver-3.
|
|
- 'singlestep_fixed':
|
|
Fixed order singlestep DPM-Solver (i.e. DPM-Solver-1 or singlestep DPM-Solver-2 or singlestep DPM-Solver-3).
|
|
We use singlestep DPM-Solver-`order` for `order`=1 or 2 or 3, with total [`steps` // `order`] * `order` NFE.
|
|
- 'adaptive':
|
|
Adaptive step size DPM-Solver (i.e. "DPM-Solver-12" and "DPM-Solver-23" in the paper).
|
|
We ignore `steps` and use adaptive step size DPM-Solver with a higher order of `order`.
|
|
You can adjust the absolute tolerance `atol` and the relative tolerance `rtol` to balance the computatation costs
|
|
(NFE) and the sample quality.
|
|
- If `order` == 2, we use DPM-Solver-12 which combines DPM-Solver-1 and singlestep DPM-Solver-2.
|
|
- If `order` == 3, we use DPM-Solver-23 which combines singlestep DPM-Solver-2 and singlestep DPM-Solver-3.
|
|
|
|
=====================================================
|
|
|
|
Some advices for choosing the algorithm:
|
|
- For **unconditional sampling** or **guided sampling with small guidance scale** by DPMs:
|
|
Use singlestep DPM-Solver or DPM-Solver++ ("DPM-Solver-fast" in the paper) with `order = 3`.
|
|
e.g., DPM-Solver:
|
|
>>> dpm_solver = DPM_Solver(model_fn, noise_schedule, algorithm_type="dpmsolver")
|
|
>>> x_sample = dpm_solver.sample(x, steps=steps, t_start=t_start, t_end=t_end, order=3,
|
|
skip_type='time_uniform', method='singlestep')
|
|
e.g., DPM-Solver++:
|
|
>>> dpm_solver = DPM_Solver(model_fn, noise_schedule, algorithm_type="dpmsolver++")
|
|
>>> x_sample = dpm_solver.sample(x, steps=steps, t_start=t_start, t_end=t_end, order=3,
|
|
skip_type='time_uniform', method='singlestep')
|
|
- For **guided sampling with large guidance scale** by DPMs:
|
|
Use multistep DPM-Solver with `algorithm_type="dpmsolver++"` and `order = 2`.
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e.g.
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>>> dpm_solver = DPM_Solver(model_fn, noise_schedule, algorithm_type="dpmsolver++")
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>>> x_sample = dpm_solver.sample(x, steps=steps, t_start=t_start, t_end=t_end, order=2,
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skip_type='time_uniform', method='multistep')
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We support three types of `skip_type`:
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- 'logSNR': uniform logSNR for the time steps. **Recommended for low-resolutional images**
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- 'time_uniform': uniform time for the time steps. **Recommended for high-resolutional images**.
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- 'time_quadratic': quadratic time for the time steps.
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=====================================================
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Args:
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x: A pytorch tensor. The initial value at time `t_start`
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e.g. if `t_start` == T, then `x` is a sample from the standard normal distribution.
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steps: A `int`. The total number of function evaluations (NFE).
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t_start: A `float`. The starting time of the sampling.
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If `T` is None, we use self.noise_schedule.T (default is 1.0).
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t_end: A `float`. The ending time of the sampling.
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If `t_end` is None, we use 1. / self.noise_schedule.total_N.
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e.g. if total_N == 1000, we have `t_end` == 1e-3.
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For discrete-time DPMs:
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- We recommend `t_end` == 1. / self.noise_schedule.total_N.
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For continuous-time DPMs:
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- We recommend `t_end` == 1e-3 when `steps` <= 15; and `t_end` == 1e-4 when `steps` > 15.
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order: A `int`. The order of DPM-Solver.
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skip_type: A `str`. The type for the spacing of the time steps. 'time_uniform' or 'logSNR' or 'time_quadratic'.
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method: A `str`. The method for sampling. 'singlestep' or 'multistep' or 'singlestep_fixed' or 'adaptive'.
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denoise_to_zero: A `bool`. Whether to denoise to time 0 at the final step.
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Default is `False`. If `denoise_to_zero` is `True`, the total NFE is (`steps` + 1).
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This trick is firstly proposed by DDPM (https://arxiv.org/abs/2006.11239) and
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score_sde (https://arxiv.org/abs/2011.13456). Such trick can improve the FID
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for diffusion models sampling by diffusion SDEs for low-resolutional images
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(such as CIFAR-10). However, we observed that such trick does not matter for
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high-resolutional images. As it needs an additional NFE, we do not recommend
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it for high-resolutional images.
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lower_order_final: A `bool`. Whether to use lower order solvers at the final steps.
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Only valid for `method=multistep` and `steps < 15`. We empirically find that
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this trick is a key to stabilizing the sampling by DPM-Solver with very few steps
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(especially for steps <= 10). So we recommend to set it to be `True`.
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solver_type: A `str`. The taylor expansion type for the solver. `dpmsolver` or `taylor`. We recommend `dpmsolver`.
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atol: A `float`. The absolute tolerance of the adaptive step size solver. Valid when `method` == 'adaptive'.
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rtol: A `float`. The relative tolerance of the adaptive step size solver. Valid when `method` == 'adaptive'.
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return_intermediate: A `bool`. Whether to save the xt at each step.
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When set to `True`, method returns a tuple (x0, intermediates); when set to False, method returns only x0.
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Returns:
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x_end: A pytorch tensor. The approximated solution at time `t_end`.
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"""
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t_0 = 1. / self.noise_schedule.total_N if t_end is None else t_end
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t_T = self.noise_schedule.T if t_start is None else t_start
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assert t_0 > 0 and t_T > 0, "Time range needs to be greater than 0. For discrete-time DPMs, it needs to be in [1 / N, 1], where N is the length of betas array"
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if return_intermediate:
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assert method in ['multistep', 'singlestep', 'singlestep_fixed'], "Cannot use adaptive solver when saving intermediate values"
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if self.correcting_xt_fn is not None:
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assert method in ['multistep', 'singlestep', 'singlestep_fixed'], "Cannot use adaptive solver when correcting_xt_fn is not None"
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device = x.device
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intermediates = []
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with torch.no_grad():
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if method == 'adaptive':
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x = self.dpm_solver_adaptive(x, order=order, t_T=t_T, t_0=t_0, atol=atol, rtol=rtol, solver_type=solver_type)
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elif method == 'multistep':
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assert steps >= order
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timesteps = self.get_time_steps(skip_type=skip_type, t_T=t_T, t_0=t_0, N=steps, device=device)
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assert timesteps.shape[0] - 1 == steps
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# Init the initial values.
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step = 0
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t = timesteps[step]
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t_prev_list = [t]
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model_prev_list = [self.model_fn(x, t)]
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if self.correcting_xt_fn is not None:
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x = self.correcting_xt_fn(x, t, step)
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if return_intermediate:
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intermediates.append(x)
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# Init the first `order` values by lower order multistep DPM-Solver.
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for step in range(1, order):
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t = timesteps[step]
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x = self.multistep_dpm_solver_update(x, model_prev_list, t_prev_list, t, step, solver_type=solver_type)
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if self.correcting_xt_fn is not None:
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x = self.correcting_xt_fn(x, t, step)
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if return_intermediate:
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intermediates.append(x)
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t_prev_list.append(t)
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model_prev_list.append(self.model_fn(x, t))
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# Compute the remaining values by `order`-th order multistep DPM-Solver.
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for step in range(order, steps + 1):
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t = timesteps[step]
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# We only use lower order for steps < 10
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if lower_order_final and steps < 10:
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step_order = min(order, steps + 1 - step)
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else:
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step_order = order
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x = self.multistep_dpm_solver_update(x, model_prev_list, t_prev_list, t, step_order, solver_type=solver_type)
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if self.correcting_xt_fn is not None:
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x = self.correcting_xt_fn(x, t, step)
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if return_intermediate:
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intermediates.append(x)
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for i in range(order - 1):
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t_prev_list[i] = t_prev_list[i + 1]
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model_prev_list[i] = model_prev_list[i + 1]
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t_prev_list[-1] = t
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# We do not need to evaluate the final model value.
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if step < steps:
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model_prev_list[-1] = self.model_fn(x, t)
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elif method in ['singlestep', 'singlestep_fixed']:
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if method == 'singlestep':
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timesteps_outer, orders = self.get_orders_and_timesteps_for_singlestep_solver(steps=steps, order=order, skip_type=skip_type, t_T=t_T, t_0=t_0, device=device)
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elif method == 'singlestep_fixed':
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K = steps // order
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orders = [order,] * K
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timesteps_outer = self.get_time_steps(skip_type=skip_type, t_T=t_T, t_0=t_0, N=K, device=device)
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for step, order in enumerate(orders):
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s, t = timesteps_outer[step], timesteps_outer[step + 1]
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timesteps_inner = self.get_time_steps(skip_type=skip_type, t_T=s.item(), t_0=t.item(), N=order, device=device)
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lambda_inner = self.noise_schedule.marginal_lambda(timesteps_inner)
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h = lambda_inner[-1] - lambda_inner[0]
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r1 = None if order <= 1 else (lambda_inner[1] - lambda_inner[0]) / h
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r2 = None if order <= 2 else (lambda_inner[2] - lambda_inner[0]) / h
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x = self.singlestep_dpm_solver_update(x, s, t, order, solver_type=solver_type, r1=r1, r2=r2)
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if self.correcting_xt_fn is not None:
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x = self.correcting_xt_fn(x, t, step)
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if return_intermediate:
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intermediates.append(x)
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else:
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raise ValueError("Got wrong method {}".format(method))
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if denoise_to_zero:
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t = torch.ones((1,)).to(device) * t_0
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x = self.denoise_to_zero_fn(x, t)
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if self.correcting_xt_fn is not None:
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x = self.correcting_xt_fn(x, t, step + 1)
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if return_intermediate:
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intermediates.append(x)
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if return_intermediate:
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return x, intermediates
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else:
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return x
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|
|
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#############################################################
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# other utility functions
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|
#############################################################
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def interpolate_fn(x, xp, yp):
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"""
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A piecewise linear function y = f(x), using xp and yp as keypoints.
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|
We implement f(x) in a differentiable way (i.e. applicable for autograd).
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|
The function f(x) is well-defined for all x-axis. (For x beyond the bounds of xp, we use the outmost points of xp to define the linear function.)
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|
|
|
Args:
|
|
x: PyTorch tensor with shape [N, C], where N is the batch size, C is the number of channels (we use C = 1 for DPM-Solver).
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|
xp: PyTorch tensor with shape [C, K], where K is the number of keypoints.
|
|
yp: PyTorch tensor with shape [C, K].
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|
Returns:
|
|
The function values f(x), with shape [N, C].
|
|
"""
|
|
N, K = x.shape[0], xp.shape[1]
|
|
all_x = torch.cat([x.unsqueeze(2), xp.unsqueeze(0).repeat((N, 1, 1))], dim=2)
|
|
sorted_all_x, x_indices = torch.sort(all_x, dim=2)
|
|
x_idx = torch.argmin(x_indices, dim=2)
|
|
cand_start_idx = x_idx - 1
|
|
start_idx = torch.where(
|
|
torch.eq(x_idx, 0),
|
|
torch.tensor(1, device=x.device),
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|
torch.where(
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|
torch.eq(x_idx, K), torch.tensor(K - 2, device=x.device), cand_start_idx,
|
|
),
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)
|
|
end_idx = torch.where(torch.eq(start_idx, cand_start_idx), start_idx + 2, start_idx + 1)
|
|
start_x = torch.gather(sorted_all_x, dim=2, index=start_idx.unsqueeze(2)).squeeze(2)
|
|
end_x = torch.gather(sorted_all_x, dim=2, index=end_idx.unsqueeze(2)).squeeze(2)
|
|
start_idx2 = torch.where(
|
|
torch.eq(x_idx, 0),
|
|
torch.tensor(0, device=x.device),
|
|
torch.where(
|
|
torch.eq(x_idx, K), torch.tensor(K - 2, device=x.device), cand_start_idx,
|
|
),
|
|
)
|
|
y_positions_expanded = yp.unsqueeze(0).expand(N, -1, -1)
|
|
start_y = torch.gather(y_positions_expanded, dim=2, index=start_idx2.unsqueeze(2)).squeeze(2)
|
|
end_y = torch.gather(y_positions_expanded, dim=2, index=(start_idx2 + 1).unsqueeze(2)).squeeze(2)
|
|
cand = start_y + (x - start_x) * (end_y - start_y) / (end_x - start_x)
|
|
return cand
|
|
|
|
|
|
def expand_dims(v, dims):
|
|
"""
|
|
Expand the tensor `v` to the dim `dims`.
|
|
|
|
Args:
|
|
`v`: a PyTorch tensor with shape [N].
|
|
`dim`: a `int`.
|
|
Returns:
|
|
a PyTorch tensor with shape [N, 1, 1, ..., 1] and the total dimension is `dims`.
|
|
"""
|
|
return v[(...,) + (None,)*(dims - 1)] |