diff options
Diffstat (limited to 'ldm/models/diffusion/dpm_solver/dpm_solver.py')
| -rw-r--r-- | ldm/models/diffusion/dpm_solver/dpm_solver.py | 1184 |
1 files changed, 1184 insertions, 0 deletions
diff --git a/ldm/models/diffusion/dpm_solver/dpm_solver.py b/ldm/models/diffusion/dpm_solver/dpm_solver.py new file mode 100644 index 00000000..bdb64e0c --- /dev/null +++ b/ldm/models/diffusion/dpm_solver/dpm_solver.py @@ -0,0 +1,1184 @@ +import torch +import torch.nn.functional as F +import math + + +class NoiseScheduleVP: + def __init__( + self, + schedule='discrete', + betas=None, + alphas_cumprod=None, + continuous_beta_0=0.1, + continuous_beta_1=20., + ): + """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(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 two types of VPSDEs: linear (DDPM) and cosine (improved-DDPM). The hyperparameters for the noise + schedule are the default settings in DDPM and improved-DDPM: + + 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. + cosine_s: A `float` number. The hyperparameter in the cosine schedule. + cosine_beta_max: A `float` number. The hyperparameter in the cosine 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' or 'cosine' 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', 'cosine']: + raise ValueError("Unsupported noise schedule {}. The schedule needs to be 'discrete' or 'linear' or 'cosine'".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.total_N = len(log_alphas) + self.T = 1. + self.t_array = torch.linspace(0., 1., self.total_N + 1)[1:].reshape((1, -1)) + self.log_alpha_array = log_alphas.reshape((1, -1,)) + else: + self.total_N = 1000 + self.beta_0 = continuous_beta_0 + self.beta_1 = continuous_beta_1 + self.cosine_s = 0.008 + self.cosine_beta_max = 999. + self.cosine_t_max = math.atan(self.cosine_beta_max * (1. + self.cosine_s) / math.pi) * 2. * (1. + self.cosine_s) / math.pi - self.cosine_s + self.cosine_log_alpha_0 = math.log(math.cos(self.cosine_s / (1. + self.cosine_s) * math.pi / 2.)) + self.schedule = schedule + if schedule == 'cosine': + # For the cosine schedule, T = 1 will have numerical issues. So we manually set the ending time T. + # Note that T = 0.9946 may be not the optimal setting. However, we find it works well. + self.T = 0.9946 + else: + self.T = 1. + + 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 + elif self.schedule == 'cosine': + log_alpha_fn = lambda s: torch.log(torch.cos((s + self.cosine_s) / (1. + self.cosine_s) * math.pi / 2.)) + log_alpha_t = log_alpha_fn(t) - self.cosine_log_alpha_0 + return log_alpha_t + + 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,)) + else: + log_alpha = -0.5 * torch.logaddexp(-2. * lamb, torch.zeros((1,)).to(lamb)) + t_fn = lambda log_alpha_t: torch.arccos(torch.exp(log_alpha_t + self.cosine_log_alpha_0)) * 2. * (1. + self.cosine_s) / math.pi - self.cosine_s + t = t_fn(log_alpha) + return t + + +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) * 1000. + else: + return t_continuous + + def noise_pred_fn(x, t_continuous, cond=None): + if t_continuous.reshape((-1,)).shape[0] == 1: + t_continuous = t_continuous.expand((x.shape[0])) + 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) + dims = x.dim() + return (x - expand_dims(alpha_t, dims) * output) / expand_dims(sigma_t, dims) + elif model_type == "v": + alpha_t, sigma_t = noise_schedule.marginal_alpha(t_continuous), noise_schedule.marginal_std(t_continuous) + dims = x.dim() + return expand_dims(alpha_t, dims) * output + expand_dims(sigma_t, dims) * x + elif model_type == "score": + sigma_t = noise_schedule.marginal_std(t_continuous) + dims = x.dim() + return -expand_dims(sigma_t, dims) * 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 t_continuous.reshape((-1,)).shape[0] == 1: + t_continuous = t_continuous.expand((x.shape[0])) + 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, dims=cond_grad.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"] + assert guidance_type in ["uncond", "classifier", "classifier-free"] + return model_fn + + +class DPM_Solver: + def __init__(self, model_fn, noise_schedule, predict_x0=False, thresholding=False, max_val=1.): + """Construct a DPM-Solver. + + We support both the noise prediction model ("predicting epsilon") and the data prediction model ("predicting x0"). + If `predict_x0` is False, we use the solver for the noise prediction model (DPM-Solver). + If `predict_x0` is True, we use the solver for the data prediction model (DPM-Solver++). + In such case, we further support the "dynamic thresholding" in [1] when `thresholding` is True. + The "dynamic thresholding" can greatly improve the sample quality for pixel-space DPMs with large guidance scales. + + 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 + `` + noise_schedule: A noise schedule object, such as NoiseScheduleVP. + predict_x0: A `bool`. If true, use the data prediction model; else, use the noise prediction model. + thresholding: A `bool`. Valid when `predict_x0` is True. Whether to use the "dynamic thresholding" in [1]. + max_val: A `float`. Valid when both `predict_x0` and `thresholding` are True. The max value for 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 = model_fn + self.noise_schedule = noise_schedule + self.predict_x0 = predict_x0 + self.thresholding = thresholding + self.max_val = max_val + + 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 thresholding). + """ + noise = self.noise_prediction_fn(x, t) + dims = x.dim() + alpha_t, sigma_t = self.noise_schedule.marginal_alpha(t), self.noise_schedule.marginal_std(t) + x0 = (x - expand_dims(sigma_t, dims) * noise) / expand_dims(alpha_t, dims) + if self.thresholding: + p = 0.995 # A hyperparameter in the paper of "Imagen" [1]. + s = torch.quantile(torch.abs(x0).reshape((x0.shape[0], -1)), p, dim=1) + s = expand_dims(torch.maximum(s, self.max_val * torch.ones_like(s).to(s.device)), dims) + x0 = torch.clamp(x0, -s, s) / s + return x0 + + def model_fn(self, x, t): + """ + Convert the model to the noise prediction model or the data prediction model. + """ + if self.predict_x0: + 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)).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 (x.shape[0],). + t: A pytorch tensor. The ending time, with the shape (x.shape[0],). + 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 + dims = x.dim() + 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.predict_x0: + phi_1 = torch.expm1(-h) + if model_s is None: + model_s = self.model_fn(x, s) + x_t = ( + expand_dims(sigma_t / sigma_s, dims) * x + - expand_dims(alpha_t * phi_1, dims) * 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 = ( + expand_dims(torch.exp(log_alpha_t - log_alpha_s), dims) * x + - expand_dims(sigma_t * phi_1, dims) * 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='dpm_solver'): + """ + 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 (x.shape[0],). + t: A pytorch tensor. The ending time, with the shape (x.shape[0],). + 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 'dpm_solver' or 'taylor'. The type for the high-order solvers. + The type slightly impacts the performance. We recommend to use 'dpm_solver' type. + Returns: + x_t: A pytorch tensor. The approximated solution at time `t`. + """ + if solver_type not in ['dpm_solver', 'taylor']: + raise ValueError("'solver_type' must be either 'dpm_solver' or 'taylor', got {}".format(solver_type)) + if r1 is None: + r1 = 0.5 + ns = self.noise_schedule + dims = x.dim() + 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.predict_x0: + 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 = ( + expand_dims(sigma_s1 / sigma_s, dims) * x + - expand_dims(alpha_s1 * phi_11, dims) * model_s + ) + model_s1 = self.model_fn(x_s1, s1) + if solver_type == 'dpm_solver': + x_t = ( + expand_dims(sigma_t / sigma_s, dims) * x + - expand_dims(alpha_t * phi_1, dims) * model_s + - (0.5 / r1) * expand_dims(alpha_t * phi_1, dims) * (model_s1 - model_s) + ) + elif solver_type == 'taylor': + x_t = ( + expand_dims(sigma_t / sigma_s, dims) * x + - expand_dims(alpha_t * phi_1, dims) * model_s + + (1. / r1) * expand_dims(alpha_t * ((torch.exp(-h) - 1.) / h + 1.), dims) * (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 = ( + expand_dims(torch.exp(log_alpha_s1 - log_alpha_s), dims) * x + - expand_dims(sigma_s1 * phi_11, dims) * model_s + ) + model_s1 = self.model_fn(x_s1, s1) + if solver_type == 'dpm_solver': + x_t = ( + expand_dims(torch.exp(log_alpha_t - log_alpha_s), dims) * x + - expand_dims(sigma_t * phi_1, dims) * model_s + - (0.5 / r1) * expand_dims(sigma_t * phi_1, dims) * (model_s1 - model_s) + ) + elif solver_type == 'taylor': + x_t = ( + expand_dims(torch.exp(log_alpha_t - log_alpha_s), dims) * x + - expand_dims(sigma_t * phi_1, dims) * model_s + - (1. / r1) * expand_dims(sigma_t * ((torch.exp(h) - 1.) / h - 1.), dims) * (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='dpm_solver'): + """ + 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 (x.shape[0],). + t: A pytorch tensor. The ending time, with the shape (x.shape[0],). + 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 'dpm_solver' or 'taylor'. The type for the high-order solvers. + The type slightly impacts the performance. We recommend to use 'dpm_solver' type. + Returns: + x_t: A pytorch tensor. The approximated solution at time `t`. + """ + if solver_type not in ['dpm_solver', 'taylor']: + raise ValueError("'solver_type' must be either 'dpm_solver' or 'taylor', got {}".format(solver_type)) + if r1 is None: + r1 = 1. / 3. + if r2 is None: + r2 = 2. / 3. + ns = self.noise_schedule + dims = x.dim() + 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.predict_x0: + 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 = ( + expand_dims(sigma_s1 / sigma_s, dims) * x + - expand_dims(alpha_s1 * phi_11, dims) * model_s + ) + model_s1 = self.model_fn(x_s1, s1) + x_s2 = ( + expand_dims(sigma_s2 / sigma_s, dims) * x + - expand_dims(alpha_s2 * phi_12, dims) * model_s + + r2 / r1 * expand_dims(alpha_s2 * phi_22, dims) * (model_s1 - model_s) + ) + model_s2 = self.model_fn(x_s2, s2) + if solver_type == 'dpm_solver': + x_t = ( + expand_dims(sigma_t / sigma_s, dims) * x + - expand_dims(alpha_t * phi_1, dims) * model_s + + (1. / r2) * expand_dims(alpha_t * phi_2, dims) * (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 = ( + expand_dims(sigma_t / sigma_s, dims) * x + - expand_dims(alpha_t * phi_1, dims) * model_s + + expand_dims(alpha_t * phi_2, dims) * D1 + - expand_dims(alpha_t * phi_3, dims) * 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 = ( + expand_dims(torch.exp(log_alpha_s1 - log_alpha_s), dims) * x + - expand_dims(sigma_s1 * phi_11, dims) * model_s + ) + model_s1 = self.model_fn(x_s1, s1) + x_s2 = ( + expand_dims(torch.exp(log_alpha_s2 - log_alpha_s), dims) * x + - expand_dims(sigma_s2 * phi_12, dims) * model_s + - r2 / r1 * expand_dims(sigma_s2 * phi_22, dims) * (model_s1 - model_s) + ) + model_s2 = self.model_fn(x_s2, s2) + if solver_type == 'dpm_solver': + x_t = ( + expand_dims(torch.exp(log_alpha_t - log_alpha_s), dims) * x + - expand_dims(sigma_t * phi_1, dims) * model_s + - (1. / r2) * expand_dims(sigma_t * phi_2, dims) * (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 = ( + expand_dims(torch.exp(log_alpha_t - log_alpha_s), dims) * x + - expand_dims(sigma_t * phi_1, dims) * model_s + - expand_dims(sigma_t * phi_2, dims) * D1 + - expand_dims(sigma_t * phi_3, dims) * 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="dpm_solver"): + """ + 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 (x.shape[0],) + t: A pytorch tensor. The ending time, with the shape (x.shape[0],). + solver_type: either 'dpm_solver' or 'taylor'. The type for the high-order solvers. + The type slightly impacts the performance. We recommend to use 'dpm_solver' type. + Returns: + x_t: A pytorch tensor. The approximated solution at time `t`. + """ + if solver_type not in ['dpm_solver', 'taylor']: + raise ValueError("'solver_type' must be either 'dpm_solver' or 'taylor', got {}".format(solver_type)) + ns = self.noise_schedule + dims = x.dim() + model_prev_1, model_prev_0 = model_prev_list + t_prev_1, t_prev_0 = t_prev_list + 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 = expand_dims(1. / r0, dims) * (model_prev_0 - model_prev_1) + if self.predict_x0: + if solver_type == 'dpm_solver': + x_t = ( + expand_dims(sigma_t / sigma_prev_0, dims) * x + - expand_dims(alpha_t * (torch.exp(-h) - 1.), dims) * model_prev_0 + - 0.5 * expand_dims(alpha_t * (torch.exp(-h) - 1.), dims) * D1_0 + ) + elif solver_type == 'taylor': + x_t = ( + expand_dims(sigma_t / sigma_prev_0, dims) * x + - expand_dims(alpha_t * (torch.exp(-h) - 1.), dims) * model_prev_0 + + expand_dims(alpha_t * ((torch.exp(-h) - 1.) / h + 1.), dims) * D1_0 + ) + else: + if solver_type == 'dpm_solver': + x_t = ( + expand_dims(torch.exp(log_alpha_t - log_alpha_prev_0), dims) * x + - expand_dims(sigma_t * (torch.exp(h) - 1.), dims) * model_prev_0 + - 0.5 * expand_dims(sigma_t * (torch.exp(h) - 1.), dims) * D1_0 + ) + elif solver_type == 'taylor': + x_t = ( + expand_dims(torch.exp(log_alpha_t - log_alpha_prev_0), dims) * x + - expand_dims(sigma_t * (torch.exp(h) - 1.), dims) * model_prev_0 + - expand_dims(sigma_t * ((torch.exp(h) - 1.) / h - 1.), dims) * D1_0 + ) + return x_t + + def multistep_dpm_solver_third_update(self, x, model_prev_list, t_prev_list, t, solver_type='dpm_solver'): + """ + 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 (x.shape[0],) + t: A pytorch tensor. The ending time, with the shape (x.shape[0],). + solver_type: either 'dpm_solver' or 'taylor'. The type for the high-order solvers. + The type slightly impacts the performance. We recommend to use 'dpm_solver' type. + Returns: + x_t: A pytorch tensor. The approximated solution at time `t`. + """ + ns = self.noise_schedule + dims = x.dim() + 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 = expand_dims(1. / r0, dims) * (model_prev_0 - model_prev_1) + D1_1 = expand_dims(1. / r1, dims) * (model_prev_1 - model_prev_2) + D1 = D1_0 + expand_dims(r0 / (r0 + r1), dims) * (D1_0 - D1_1) + D2 = expand_dims(1. / (r0 + r1), dims) * (D1_0 - D1_1) + if self.predict_x0: + x_t = ( + expand_dims(sigma_t / sigma_prev_0, dims) * x + - expand_dims(alpha_t * (torch.exp(-h) - 1.), dims) * model_prev_0 + + expand_dims(alpha_t * ((torch.exp(-h) - 1.) / h + 1.), dims) * D1 + - expand_dims(alpha_t * ((torch.exp(-h) - 1. + h) / h**2 - 0.5), dims) * D2 + ) + else: + x_t = ( + expand_dims(torch.exp(log_alpha_t - log_alpha_prev_0), dims) * x + - expand_dims(sigma_t * (torch.exp(h) - 1.), dims) * model_prev_0 + - expand_dims(sigma_t * ((torch.exp(h) - 1.) / h - 1.), dims) * D1 + - expand_dims(sigma_t * ((torch.exp(h) - 1. - h) / h**2 - 0.5), dims) * D2 + ) + return x_t + + def singlestep_dpm_solver_update(self, x, s, t, order, return_intermediate=False, solver_type='dpm_solver', 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 (x.shape[0],). + t: A pytorch tensor. The ending time, with the shape (x.shape[0],). + 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 'dpm_solver' or 'taylor'. The type for the high-order solvers. + The type slightly impacts the performance. We recommend to use 'dpm_solver' 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='dpm_solver'): + """ + 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 (x.shape[0],) + t: A pytorch tensor. The ending time, with the shape (x.shape[0],). + order: A `int`. The order of DPM-Solver. We only support order == 1 or 2 or 3. + solver_type: either 'dpm_solver' or 'taylor'. The type for the high-order solvers. + The type slightly impacts the performance. We recommend to use 'dpm_solver' 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='dpm_solver'): + """ + 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 'dpm_solver' or 'taylor'. The type for the high-order solvers. + The type slightly impacts the performance. We recommend to use 'dpm_solver' 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((x.shape[0],)).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 + lower_update = lambda x, s, t: self.dpm_solver_first_update(x, s, t, return_intermediate=True) + higher_update = lambda x, s, t, **kwargs: 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. + lower_update = lambda x, s, t: self.singlestep_dpm_solver_second_update(x, s, t, r1=r1, return_intermediate=True, solver_type=solver_type) + higher_update = lambda x, s, t, **kwargs: 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))) + norm_fn = lambda v: 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 sample(self, x, steps=20, t_start=None, t_end=None, order=3, skip_type='time_uniform', + method='singlestep', lower_order_final=True, denoise_to_zero=False, solver_type='dpm_solver', + atol=0.0078, rtol=0.05, + ): + """ + 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 ("DPM-Solver-fast" in the paper) with `order = 3`. + e.g. + >>> dpm_solver = DPM_Solver(model_fn, noise_schedule, predict_x0=False) + >>> 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 `predict_x0 = True` and `order = 2`. + e.g. + >>> dpm_solver = DPM_Solver(model_fn, noise_schedule, predict_x0=True) + >>> 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. `dpm_solver` or `taylor`. We recommend `dpm_solver`. + 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'. + 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 + device = x.device + if method == 'adaptive': + with torch.no_grad(): + 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 + with torch.no_grad(): + vec_t = timesteps[0].expand((x.shape[0])) + model_prev_list = [self.model_fn(x, vec_t)] + t_prev_list = [vec_t] + # Init the first `order` values by lower order multistep DPM-Solver. + for init_order in range(1, order): + vec_t = timesteps[init_order].expand(x.shape[0]) + x = self.multistep_dpm_solver_update(x, model_prev_list, t_prev_list, vec_t, init_order, solver_type=solver_type) + model_prev_list.append(self.model_fn(x, vec_t)) + t_prev_list.append(vec_t) + # Compute the remaining values by `order`-th order multistep DPM-Solver. + for step in range(order, steps + 1): + vec_t = timesteps[step].expand(x.shape[0]) + if lower_order_final and steps < 15: + step_order = min(order, steps + 1 - step) + else: + step_order = order + x = self.multistep_dpm_solver_update(x, model_prev_list, t_prev_list, vec_t, step_order, solver_type=solver_type) + 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] = vec_t + # We do not need to evaluate the final model value. + if step < steps: + model_prev_list[-1] = self.model_fn(x, vec_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 i, order in enumerate(orders): + t_T_inner, t_0_inner = timesteps_outer[i], timesteps_outer[i + 1] + timesteps_inner = self.get_time_steps(skip_type=skip_type, t_T=t_T_inner.item(), t_0=t_0_inner.item(), N=order, device=device) + lambda_inner = self.noise_schedule.marginal_lambda(timesteps_inner) + vec_s, vec_t = t_T_inner.tile(x.shape[0]), t_0_inner.tile(x.shape[0]) + 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, vec_s, vec_t, order, solver_type=solver_type, r1=r1, r2=r2) + if denoise_to_zero: + x = self.denoise_to_zero_fn(x, torch.ones((x.shape[0],)).to(device) * t_0) + 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)]
\ No newline at end of file |