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| import torch | |
| class NoiseScheduleVP: | |
| def __init__( | |
| self, | |
| schedule='discrete', | |
| betas=None, | |
| alphas_cumprod=None, | |
| continuous_beta_0=0.1, | |
| continuous_beta_1=20., | |
| dtype=torch.float32, | |
| ): | |
| """Create a wrapper class for the forward SDE (VP type). | |
| *** | |
| Update: We support discrete-time diffusion models by implementing a picewise linear interpolation for log_alpha_t. | |
| We recommend to use schedule='discrete' for the discrete-time diffusion models, especially for high-resolution images. | |
| *** | |
| The forward SDE ensures that the condition distribution q_{t|0}(x_t | x_0) = N ( alpha_t * x_0, sigma_t^2 * I ). | |
| We further define lambda_t = log(alpha_t) - log(sigma_t), which is the half-logSNR (described in the DPM-Solver paper). | |
| Therefore, we implement the functions for computing alpha_t, sigma_t and lambda_t. For t in [0, T], we have: | |
| log_alpha_t = self.marginal_log_mean_coeff(t) | |
| sigma_t = self.marginal_std(t) | |
| lambda_t = self.marginal_lambda(t) | |
| Moreover, as lambda(t) is an invertible function, we also support its inverse function: | |
| t = self.inverse_lambda(lambda_t) | |
| =============================================================== | |
| We support both discrete-time DPMs (trained on n = 0, 1, ..., N-1) and continuous-time DPMs (trained on t in [t_0, T]). | |
| 1. For discrete-time DPMs: | |
| For discrete-time DPMs trained on n = 0, 1, ..., N-1, we convert the discrete steps to continuous time steps by: | |
| t_i = (i + 1) / N | |
| e.g. for N = 1000, we have t_0 = 1e-3 and T = t_{N-1} = 1. | |
| We solve the corresponding diffusion ODE from time T = 1 to time t_0 = 1e-3. | |
| Args: | |
| betas: A `torch.Tensor`. The beta array for the discrete-time DPM. (See the original DDPM paper for details) | |
| alphas_cumprod: A `torch.Tensor`. The cumprod alphas for the discrete-time DPM. (See the original DDPM paper for details) | |
| Note that we always have alphas_cumprod = cumprod(1 - betas). Therefore, we only need to set one of `betas` and `alphas_cumprod`. | |
| **Important**: Please pay special attention for the args for `alphas_cumprod`: | |
| The `alphas_cumprod` is the \hat{alpha_n} arrays in the notations of DDPM. Specifically, DDPMs assume that | |
| q_{t_n | 0}(x_{t_n} | x_0) = N ( \sqrt{\hat{alpha_n}} * x_0, (1 - \hat{alpha_n}) * I ). | |
| Therefore, the notation \hat{alpha_n} is different from the notation alpha_t in DPM-Solver. In fact, we have | |
| alpha_{t_n} = \sqrt{\hat{alpha_n}}, | |
| and | |
| log(alpha_{t_n}) = 0.5 * log(\hat{alpha_n}). | |
| 2. For continuous-time DPMs: | |
| We support the linear VPSDE for the continuous time setting. The hyperparameters for the noise | |
| schedule are the default settings in Yang Song's ScoreSDE: | |
| Args: | |
| beta_min: A `float` number. The smallest beta for the linear schedule. | |
| beta_max: A `float` number. The largest beta for the linear schedule. | |
| T: A `float` number. The ending time of the forward process. | |
| =============================================================== | |
| Args: | |
| schedule: A `str`. The noise schedule of the forward SDE. 'discrete' for discrete-time DPMs, | |
| 'linear' for continuous-time DPMs. | |
| Returns: | |
| A wrapper object of the forward SDE (VP type). | |
| =============================================================== | |
| Example: | |
| # For discrete-time DPMs, given betas (the beta array for n = 0, 1, ..., N - 1): | |
| >>> ns = NoiseScheduleVP('discrete', betas=betas) | |
| # For discrete-time DPMs, given alphas_cumprod (the \hat{alpha_n} array for n = 0, 1, ..., N - 1): | |
| >>> ns = NoiseScheduleVP('discrete', alphas_cumprod=alphas_cumprod) | |
| # For continuous-time DPMs (VPSDE), linear schedule: | |
| >>> ns = NoiseScheduleVP('linear', continuous_beta_0=0.1, continuous_beta_1=20.) | |
| """ | |
| if schedule not in ['discrete', 'linear']: | |
| raise ValueError("Unsupported noise schedule {}. The schedule needs to be 'discrete' or 'linear'".format(schedule)) | |
| self.schedule = schedule | |
| if schedule == 'discrete': | |
| if betas is not None: | |
| log_alphas = 0.5 * torch.log(1 - betas).cumsum(dim=0) | |
| else: | |
| assert alphas_cumprod is not None | |
| log_alphas = 0.5 * torch.log(alphas_cumprod) | |
| self.T = 1. | |
| self.log_alpha_array = self.numerical_clip_alpha(log_alphas).reshape((1, -1,)).to(dtype=dtype) | |
| self.total_N = self.log_alpha_array.shape[1] | |
| self.t_array = torch.linspace(0., 1., self.total_N + 1)[1:].reshape((1, -1)).to(dtype=dtype) | |
| else: | |
| self.T = 1. | |
| self.total_N = 1000 | |
| self.beta_0 = continuous_beta_0 | |
| self.beta_1 = continuous_beta_1 | |
| def numerical_clip_alpha(self, log_alphas, clipped_lambda=-5.1): | |
| """ | |
| For some beta schedules such as cosine schedule, the log-SNR has numerical isssues. | |
| We clip the log-SNR near t=T within -5.1 to ensure the stability. | |
| Such a trick is very useful for diffusion models with the cosine schedule, such as i-DDPM, guided-diffusion and GLIDE. | |
| """ | |
| log_sigmas = 0.5 * torch.log(1. - torch.exp(2. * log_alphas)) | |
| lambs = log_alphas - log_sigmas | |
| idx = torch.searchsorted(torch.flip(lambs, [0]), clipped_lambda) | |
| if idx > 0: | |
| log_alphas = log_alphas[:-idx] | |
| return log_alphas | |
| def marginal_log_mean_coeff(self, t): | |
| """ | |
| Compute log(alpha_t) of a given continuous-time label t in [0, T]. | |
| """ | |
| if self.schedule == 'discrete': | |
| return interpolate_fn(t.reshape((-1, 1)), self.t_array.to(t.device), self.log_alpha_array.to(t.device)).reshape((-1)) | |
| elif self.schedule == 'linear': | |
| return -0.25 * t ** 2 * (self.beta_1 - self.beta_0) - 0.5 * t * self.beta_0 | |
| def marginal_alpha(self, t): | |
| """ | |
| Compute alpha_t of a given continuous-time label t in [0, T]. | |
| """ | |
| return torch.exp(self.marginal_log_mean_coeff(t)) | |
| def marginal_std(self, t): | |
| """ | |
| Compute sigma_t of a given continuous-time label t in [0, T]. | |
| """ | |
| return torch.sqrt(1. - torch.exp(2. * self.marginal_log_mean_coeff(t))) | |
| def marginal_lambda(self, t): | |
| """ | |
| Compute lambda_t = log(alpha_t) - log(sigma_t) of a given continuous-time label t in [0, T]. | |
| """ | |
| log_mean_coeff = self.marginal_log_mean_coeff(t) | |
| log_std = 0.5 * torch.log(1. - torch.exp(2. * log_mean_coeff)) | |
| return log_mean_coeff - log_std | |
| def inverse_lambda(self, lamb): | |
| """ | |
| Compute the continuous-time label t in [0, T] of a given half-logSNR lambda_t. | |
| """ | |
| if self.schedule == 'linear': | |
| tmp = 2. * (self.beta_1 - self.beta_0) * torch.logaddexp(-2. * lamb, torch.zeros((1,)).to(lamb)) | |
| Delta = self.beta_0**2 + tmp | |
| return tmp / (torch.sqrt(Delta) + self.beta_0) / (self.beta_1 - self.beta_0) | |
| elif self.schedule == 'discrete': | |
| log_alpha = -0.5 * torch.logaddexp(torch.zeros((1,)).to(lamb.device), -2. * lamb) | |
| 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])) | |
| return t.reshape((-1,)) | |
| def model_wrapper( | |
| model, | |
| noise_schedule, | |
| model_type="noise", | |
| model_kwargs={}, | |
| guidance_type="uncond", | |
| condition=None, | |
| unconditional_condition=None, | |
| guidance_scale=1., | |
| classifier_fn=None, | |
| classifier_kwargs={}, | |
| ): | |
| """Create a wrapper function for the noise prediction model. | |
| DPM-Solver needs to solve the continuous-time diffusion ODEs. For DPMs trained on discrete-time labels, we need to | |
| firstly wrap the model function to a noise prediction model that accepts the continuous time as the input. | |
| We support four types of the diffusion model by setting `model_type`: | |
| 1. "noise": noise prediction model. (Trained by predicting noise). | |
| 2. "x_start": data prediction model. (Trained by predicting the data x_0 at time 0). | |
| 3. "v": velocity prediction model. (Trained by predicting the velocity). | |
| The "v" prediction is derivation detailed in Appendix D of [1], and is used in Imagen-Video [2]. | |
| [1] Salimans, Tim, and Jonathan Ho. "Progressive distillation for fast sampling of diffusion models." | |
| arXiv preprint arXiv:2202.00512 (2022). | |
| [2] Ho, Jonathan, et al. "Imagen Video: High Definition Video Generation with Diffusion Models." | |
| arXiv preprint arXiv:2210.02303 (2022). | |
| 4. "score": marginal score function. (Trained by denoising score matching). | |
| Note that the score function and the noise prediction model follows a simple relationship: | |
| ``` | |
| noise(x_t, t) = -sigma_t * score(x_t, t) | |
| ``` | |
| We support three types of guided sampling by DPMs by setting `guidance_type`: | |
| 1. "uncond": unconditional sampling by DPMs. | |
| The input `model` has the following format: | |
| `` | |
| model(x, t_input, **model_kwargs) -> noise | x_start | v | score | |
| `` | |
| 2. "classifier": classifier guidance sampling [3] by DPMs and another classifier. | |
| The input `model` has the following format: | |
| `` | |
| model(x, t_input, **model_kwargs) -> noise | x_start | v | score | |
| `` | |
| The input `classifier_fn` has the following format: | |
| `` | |
| classifier_fn(x, t_input, cond, **classifier_kwargs) -> logits(x, t_input, cond) | |
| `` | |
| [3] P. Dhariwal and A. Q. Nichol, "Diffusion models beat GANs on image synthesis," | |
| in Advances in Neural Information Processing Systems, vol. 34, 2021, pp. 8780-8794. | |
| 3. "classifier-free": classifier-free guidance sampling by conditional DPMs. | |
| The input `model` has the following format: | |
| `` | |
| model(x, t_input, cond, **model_kwargs) -> noise | x_start | v | score | |
| `` | |
| And if cond == `unconditional_condition`, the model output is the unconditional DPM output. | |
| [4] Ho, Jonathan, and Tim Salimans. "Classifier-free diffusion guidance." | |
| arXiv preprint arXiv:2207.12598 (2022). | |
| The `t_input` is the time label of the model, which may be discrete-time labels (i.e. 0 to 999) | |
| or continuous-time labels (i.e. epsilon to T). | |
| We wrap the model function to accept only `x` and `t_continuous` as inputs, and outputs the predicted noise: | |
| `` | |
| def model_fn(x, t_continuous) -> noise: | |
| t_input = get_model_input_time(t_continuous) | |
| return noise_pred(model, x, t_input, **model_kwargs) | |
| `` | |
| where `t_continuous` is the continuous time labels (i.e. epsilon to T). And we use `model_fn` for DPM-Solver. | |
| =============================================================== | |
| Args: | |
| model: A diffusion model with the corresponding format described above. | |
| noise_schedule: A noise schedule object, such as NoiseScheduleVP. | |
| model_type: A `str`. The parameterization type of the diffusion model. | |
| "noise" or "x_start" or "v" or "score". | |
| model_kwargs: A `dict`. A dict for the other inputs of the model function. | |
| guidance_type: A `str`. The type of the guidance for sampling. | |
| "uncond" or "classifier" or "classifier-free". | |
| condition: A pytorch tensor. The condition for the guided sampling. | |
| Only used for "classifier" or "classifier-free" guidance type. | |
| unconditional_condition: A pytorch tensor. The condition for the unconditional sampling. | |
| Only used for "classifier-free" guidance type. | |
| guidance_scale: A `float`. The scale for the guided sampling. | |
| classifier_fn: A classifier function. Only used for the classifier guidance. | |
| classifier_kwargs: A `dict`. A dict for the other inputs of the classifier function. | |
| Returns: | |
| A noise prediction model that accepts the noised data and the continuous time as the inputs. | |
| """ | |
| def get_model_input_time(t_continuous): | |
| """ | |
| Convert the continuous-time `t_continuous` (in [epsilon, T]) to the model input time. | |
| For discrete-time DPMs, we convert `t_continuous` in [1 / N, 1] to `t_input` in [0, 1000 * (N - 1) / N]. | |
| For continuous-time DPMs, we just use `t_continuous`. | |
| """ | |
| if noise_schedule.schedule == 'discrete': | |
| return (t_continuous - 1. / noise_schedule.total_N) * noise_schedule.total_N | |
| else: | |
| return t_continuous | |
| def noise_pred_fn(x, t_continuous, cond=None): | |
| t_input = get_model_input_time(t_continuous) | |
| if cond is None: | |
| output = model(x, t_input, **model_kwargs) | |
| else: | |
| output = model(x, t_input, cond, **model_kwargs) | |
| if model_type == "noise": | |
| return output | |
| elif model_type == "x_start": | |
| alpha_t, sigma_t = noise_schedule.marginal_alpha(t_continuous), noise_schedule.marginal_std(t_continuous) | |
| return (x - expand_dims(alpha_t, x.dim()) * output) / expand_dims(sigma_t, x.dim()) | |
| elif model_type == "v": | |
| alpha_t, sigma_t = noise_schedule.marginal_alpha(t_continuous), noise_schedule.marginal_std(t_continuous) | |
| return expand_dims(alpha_t, x.dim()) * output + expand_dims(sigma_t, x.dim()) * x | |
| elif model_type == "score": | |
| sigma_t = noise_schedule.marginal_std(t_continuous) | |
| return -expand_dims(sigma_t, x.dim()) * output | |
| def cond_grad_fn(x, t_input): | |
| """ | |
| Compute the gradient of the classifier, i.e. nabla_{x} log p_t(cond | x_t). | |
| """ | |
| with torch.enable_grad(): | |
| x_in = x.detach().requires_grad_(True) | |
| log_prob = classifier_fn(x_in, t_input, condition, **classifier_kwargs) | |
| return torch.autograd.grad(log_prob.sum(), x_in)[0] | |
| def model_fn(x, t_continuous): | |
| """ | |
| The noise predicition model function that is used for DPM-Solver. | |
| """ | |
| if guidance_type == "uncond": | |
| return noise_pred_fn(x, t_continuous) | |
| elif guidance_type == "classifier": | |
| assert classifier_fn is not None | |
| t_input = get_model_input_time(t_continuous) | |
| cond_grad = cond_grad_fn(x, t_input) | |
| sigma_t = noise_schedule.marginal_std(t_continuous) | |
| noise = noise_pred_fn(x, t_continuous) | |
| return noise - guidance_scale * expand_dims(sigma_t, x.dim()) * cond_grad | |
| elif guidance_type == "classifier-free": | |
| if guidance_scale == 1. or unconditional_condition is None: | |
| return noise_pred_fn(x, t_continuous, cond=condition) | |
| else: | |
| x_in = torch.cat([x] * 2) | |
| t_in = torch.cat([t_continuous] * 2) | |
| c_in = torch.cat([unconditional_condition, condition]) | |
| noise_uncond, noise = noise_pred_fn(x_in, t_in, cond=c_in).chunk(2) | |
| return noise_uncond + guidance_scale * (noise - noise_uncond) | |
| assert model_type in ["noise", "x_start", "v", "score"] | |
| assert guidance_type in ["uncond", "classifier", "classifier-free"] | |
| return model_fn | |
| class DPM_Solver: | |
| def __init__( | |
| self, | |
| model_fn, | |
| noise_schedule, | |
| algorithm_type="dpmsolver++", | |
| correcting_x0_fn=None, | |
| correcting_xt_fn=None, | |
| thresholding_max_val=1., | |
| dynamic_thresholding_ratio=0.995, | |
| ): | |
| """Construct a DPM-Solver. | |
| We support both DPM-Solver (`algorithm_type="dpmsolver"`) and DPM-Solver++ (`algorithm_type="dpmsolver++"`). | |
| We also support the "dynamic thresholding" method in Imagen[1]. For pixel-space diffusion models, you | |
| can set both `algorithm_type="dpmsolver++"` and `correcting_x0_fn="dynamic_thresholding"` to use the | |
| dynamic thresholding. The "dynamic thresholding" can greatly improve the sample quality for pixel-space | |
| DPMs with large guidance scales. Note that the thresholding method is **unsuitable** for latent-space | |
| DPMs (such as stable-diffusion). | |
| To support advanced algorithms in image-to-image applications, we also support corrector functions for | |
| both x0 and xt. | |
| Args: | |
| model_fn: A noise prediction model function which accepts the continuous-time input (t in [epsilon, T]): | |
| `` | |
| def model_fn(x, t_continuous): | |
| return noise | |
| `` | |
| The shape of `x` is `(batch_size, **shape)`, and the shape of `t_continuous` is `(batch_size,)`. | |
| noise_schedule: A noise schedule object, such as NoiseScheduleVP. | |
| algorithm_type: A `str`. Either "dpmsolver" or "dpmsolver++". | |
| correcting_x0_fn: A `str` or a function with the following format: | |
| ``` | |
| def correcting_x0_fn(x0, t): | |
| x0_new = ... | |
| return x0_new | |
| ``` | |
| This function is to correct the outputs of the data prediction model at each sampling step. e.g., | |
| ``` | |
| x0_pred = data_pred_model(xt, t) | |
| if correcting_x0_fn is not None: | |
| x0_pred = correcting_x0_fn(x0_pred, t) | |
| xt_1 = update(x0_pred, xt, t) | |
| ``` | |
| If `correcting_x0_fn="dynamic_thresholding"`, we use the dynamic thresholding proposed in Imagen[1]. | |
| correcting_xt_fn: A function with the following format: | |
| ``` | |
| def correcting_xt_fn(xt, t, step): | |
| x_new = ... | |
| return x_new | |
| ``` | |
| This function is to correct the intermediate samples xt at each sampling step. e.g., | |
| ``` | |
| xt = ... | |
| xt = correcting_xt_fn(xt, t, step) | |
| ``` | |
| thresholding_max_val: A `float`. The max value for thresholding. | |
| Valid only when use `dpmsolver++` and `correcting_x0_fn="dynamic_thresholding"`. | |
| dynamic_thresholding_ratio: A `float`. The ratio for dynamic thresholding (see Imagen[1] for details). | |
| Valid only when use `dpmsolver++` and `correcting_x0_fn="dynamic_thresholding"`. | |
| [1] Chitwan Saharia, William Chan, Saurabh Saxena, Lala Li, Jay Whang, Emily Denton, Seyed Kamyar Seyed Ghasemipour, | |
| Burcu Karagol Ayan, S Sara Mahdavi, Rapha Gontijo Lopes, et al. Photorealistic text-to-image diffusion models | |
| with deep language understanding. arXiv preprint arXiv:2205.11487, 2022b. | |
| """ | |
| self.model = lambda x, t: model_fn(x, t.expand((x.shape[0]))) | |
| self.noise_schedule = noise_schedule | |
| assert algorithm_type in ["dpmsolver", "dpmsolver++"] | |
| self.algorithm_type = algorithm_type | |
| if correcting_x0_fn == "dynamic_thresholding": | |
| self.correcting_x0_fn = self.dynamic_thresholding_fn | |
| else: | |
| self.correcting_x0_fn = correcting_x0_fn | |
| self.correcting_xt_fn = correcting_xt_fn | |
| self.dynamic_thresholding_ratio = dynamic_thresholding_ratio | |
| 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`. | |
| e.g. | |
| >>> 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=2, | |
| skip_type='time_uniform', method='multistep') | |
| We support three types of `skip_type`: | |
| - 'logSNR': uniform logSNR for the time steps. **Recommended for low-resolutional images** | |
| - 'time_uniform': uniform time for the time steps. **Recommended for high-resolutional images**. | |
| - 'time_quadratic': quadratic time for the time steps. | |
| ===================================================== | |
| Args: | |
| x: A pytorch tensor. The initial value at time `t_start` | |
| e.g. if `t_start` == T, then `x` is a sample from the standard normal distribution. | |
| steps: A `int`. The total number of function evaluations (NFE). | |
| t_start: A `float`. The starting time of the sampling. | |
| If `T` is None, we use self.noise_schedule.T (default is 1.0). | |
| t_end: A `float`. The ending time of the sampling. | |
| If `t_end` is None, we use 1. / self.noise_schedule.total_N. | |
| e.g. if total_N == 1000, we have `t_end` == 1e-3. | |
| For discrete-time DPMs: | |
| - We recommend `t_end` == 1. / self.noise_schedule.total_N. | |
| For continuous-time DPMs: | |
| - We recommend `t_end` == 1e-3 when `steps` <= 15; and `t_end` == 1e-4 when `steps` > 15. | |
| order: A `int`. The order of DPM-Solver. | |
| skip_type: A `str`. The type for the spacing of the time steps. 'time_uniform' or 'logSNR' or 'time_quadratic'. | |
| method: A `str`. The method for sampling. 'singlestep' or 'multistep' or 'singlestep_fixed' or 'adaptive'. | |
| denoise_to_zero: A `bool`. Whether to denoise to time 0 at the final step. | |
| Default is `False`. If `denoise_to_zero` is `True`, the total NFE is (`steps` + 1). | |
| This trick is firstly proposed by DDPM (https://arxiv.org/abs/2006.11239) and | |
| score_sde (https://arxiv.org/abs/2011.13456). Such trick can improve the FID | |
| for diffusion models sampling by diffusion SDEs for low-resolutional images | |
| (such as CIFAR-10). However, we observed that such trick does not matter for | |
| high-resolutional images. As it needs an additional NFE, we do not recommend | |
| it for high-resolutional images. | |
| lower_order_final: A `bool`. Whether to use lower order solvers at the final steps. | |
| Only valid for `method=multistep` and `steps < 15`. We empirically find that | |
| this trick is a key to stabilizing the sampling by DPM-Solver with very few steps | |
| (especially for steps <= 10). So we recommend to set it to be `True`. | |
| solver_type: A `str`. The taylor expansion type for the solver. `dpmsolver` or `taylor`. We recommend `dpmsolver`. | |
| atol: A `float`. The absolute tolerance of the adaptive step size solver. Valid when `method` == 'adaptive'. | |
| rtol: A `float`. The relative tolerance of the adaptive step size solver. Valid when `method` == 'adaptive'. | |
| return_intermediate: A `bool`. Whether to save the xt at each step. | |
| When set to `True`, method returns a tuple (x0, intermediates); when set to False, method returns only x0. | |
| Returns: | |
| x_end: A pytorch tensor. The approximated solution at time `t_end`. | |
| """ | |
| t_0 = 1. / self.noise_schedule.total_N if t_end is None else t_end | |
| t_T = self.noise_schedule.T if t_start is None else t_start | |
| 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" | |
| if return_intermediate: | |
| assert method in ['multistep', 'singlestep', 'singlestep_fixed'], "Cannot use adaptive solver when saving intermediate values" | |
| if self.correcting_xt_fn is not None: | |
| assert method in ['multistep', 'singlestep', 'singlestep_fixed'], "Cannot use adaptive solver when correcting_xt_fn is not None" | |
| device = x.device | |
| intermediates = [] | |
| with torch.no_grad(): | |
| if method == 'adaptive': | |
| x = self.dpm_solver_adaptive(x, order=order, t_T=t_T, t_0=t_0, atol=atol, rtol=rtol, solver_type=solver_type) | |
| elif method == 'multistep': | |
| assert steps >= order | |
| timesteps = self.get_time_steps(skip_type=skip_type, t_T=t_T, t_0=t_0, N=steps, device=device) | |
| assert timesteps.shape[0] - 1 == steps | |
| # Init the initial values. | |
| step = 0 | |
| t = timesteps[step] | |
| t_prev_list = [t] | |
| model_prev_list = [self.model_fn(x, t)] | |
| if self.correcting_xt_fn is not None: | |
| x = self.correcting_xt_fn(x, t, step) | |
| if return_intermediate: | |
| intermediates.append(x) | |
| # Init the first `order` values by lower order multistep DPM-Solver. | |
| for step in range(1, order): | |
| t = timesteps[step] | |
| x = self.multistep_dpm_solver_update(x, model_prev_list, t_prev_list, t, step, solver_type=solver_type) | |
| if self.correcting_xt_fn is not None: | |
| x = self.correcting_xt_fn(x, t, step) | |
| if return_intermediate: | |
| intermediates.append(x) | |
| t_prev_list.append(t) | |
| model_prev_list.append(self.model_fn(x, t)) | |
| # Compute the remaining values by `order`-th order multistep DPM-Solver. | |
| for step in range(order, steps + 1): | |
| t = timesteps[step] | |
| # We only use lower order for steps < 10 | |
| if lower_order_final and steps < 10: | |
| step_order = min(order, steps + 1 - step) | |
| else: | |
| step_order = order | |
| x = self.multistep_dpm_solver_update(x, model_prev_list, t_prev_list, t, step_order, solver_type=solver_type) | |
| if self.correcting_xt_fn is not None: | |
| x = self.correcting_xt_fn(x, t, step) | |
| if return_intermediate: | |
| intermediates.append(x) | |
| for i in range(order - 1): | |
| t_prev_list[i] = t_prev_list[i + 1] | |
| model_prev_list[i] = model_prev_list[i + 1] | |
| t_prev_list[-1] = t | |
| # We do not need to evaluate the final model value. | |
| if step < steps: | |
| model_prev_list[-1] = self.model_fn(x, t) | |
| elif method in ['singlestep', 'singlestep_fixed']: | |
| if method == 'singlestep': | |
| 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) | |
| elif method == 'singlestep_fixed': | |
| K = steps // order | |
| orders = [order,] * K | |
| timesteps_outer = self.get_time_steps(skip_type=skip_type, t_T=t_T, t_0=t_0, N=K, device=device) | |
| for step, order in enumerate(orders): | |
| s, t = timesteps_outer[step], timesteps_outer[step + 1] | |
| timesteps_inner = self.get_time_steps(skip_type=skip_type, t_T=s.item(), t_0=t.item(), N=order, device=device) | |
| lambda_inner = self.noise_schedule.marginal_lambda(timesteps_inner) | |
| h = lambda_inner[-1] - lambda_inner[0] | |
| r1 = None if order <= 1 else (lambda_inner[1] - lambda_inner[0]) / h | |
| r2 = None if order <= 2 else (lambda_inner[2] - lambda_inner[0]) / h | |
| x = self.singlestep_dpm_solver_update(x, s, t, order, solver_type=solver_type, r1=r1, r2=r2) | |
| if self.correcting_xt_fn is not None: | |
| x = self.correcting_xt_fn(x, t, step) | |
| if return_intermediate: | |
| intermediates.append(x) | |
| else: | |
| raise ValueError("Got wrong method {}".format(method)) | |
| if denoise_to_zero: | |
| t = torch.ones((1,)).to(device) * t_0 | |
| x = self.denoise_to_zero_fn(x, t) | |
| if self.correcting_xt_fn is not None: | |
| x = self.correcting_xt_fn(x, t, step + 1) | |
| if return_intermediate: | |
| intermediates.append(x) | |
| if return_intermediate: | |
| return x, intermediates | |
| else: | |
| return x | |
| ############################################################# | |
| # other utility functions | |
| ############################################################# | |
| def interpolate_fn(x, xp, yp): | |
| """ | |
| A piecewise linear function y = f(x), using xp and yp as keypoints. | |
| We implement f(x) in a differentiable way (i.e. applicable for autograd). | |
| 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.) | |
| 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). | |
| xp: PyTorch tensor with shape [C, K], where K is the number of keypoints. | |
| yp: PyTorch tensor with shape [C, K]. | |
| 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), | |
| torch.where( | |
| torch.eq(x_idx, K), torch.tensor(K - 2, device=x.device), cand_start_idx, | |
| ), | |
| ) | |
| 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)] |