import torch import math import tqdm 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(f"Unsupported noise schedule {schedule}. The schedule needs to be 'discrete' or 'linear' or 'cosine'") 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=None, guidance_type="uncond", #condition=None, #unconditional_condition=None, guidance_scale=1., classifier_fn=None, classifier_kwargs=None, ): """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. """ model_kwargs = model_kwargs or {} classifier_kwargs = classifier_kwargs or {} 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, None, **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, condition): """ 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, condition, unconditional_condition): """ 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, condition) 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) if isinstance(condition, dict): assert isinstance(unconditional_condition, dict) c_in = {} for k in condition: if isinstance(condition[k], list): c_in[k] = [torch.cat([ unconditional_condition[k][i], condition[k][i]]) for i in range(len(condition[k]))] else: c_in[k] = torch.cat([ unconditional_condition[k], condition[k]]) elif isinstance(condition, list): c_in = [] assert isinstance(unconditional_condition, list) for i in range(len(condition)): c_in.append(torch.cat([unconditional_condition[i], condition[i]])) else: 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 UniPC: def __init__( self, model_fn, noise_schedule, predict_x0=True, thresholding=False, max_val=1., variant='bh1', condition=None, unconditional_condition=None, before_sample=None, after_sample=None, after_update=None ): """Construct a UniPC. We support both data_prediction and noise_prediction. """ self.model_fn_ = model_fn self.noise_schedule = noise_schedule self.variant = variant self.predict_x0 = predict_x0 self.thresholding = thresholding self.max_val = max_val self.condition = condition self.unconditional_condition = unconditional_condition self.before_sample = before_sample self.after_sample = after_sample self.after_update = after_update def dynamic_thresholding_fn(self, x0, t=None): """ 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 model(self, x, t): cond = self.condition uncond = self.unconditional_condition if self.before_sample is not None: x, t, cond, uncond = self.before_sample(x, t, cond, uncond) res = self.model_fn_(x, t, cond, uncond) if self.after_sample is not None: x, t, cond, uncond, res = self.after_sample(x, t, cond, uncond, res) if isinstance(res, tuple): # (None, pred_x0) res = res[1] return res 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. """ 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(f"Unsupported skip_type {skip_type}, need to be 'logSNR' or 'time_uniform' or 'time_quadratic'") 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. """ 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 = steps 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 multistep_uni_pc_update(self, x, model_prev_list, t_prev_list, t, order, **kwargs): if len(t.shape) == 0: t = t.view(-1) if 'bh' in self.variant: return self.multistep_uni_pc_bh_update(x, model_prev_list, t_prev_list, t, order, **kwargs) else: assert self.variant == 'vary_coeff' return self.multistep_uni_pc_vary_update(x, model_prev_list, t_prev_list, t, order, **kwargs) def multistep_uni_pc_vary_update(self, x, model_prev_list, t_prev_list, t, order, use_corrector=True): #print(f'using unified predictor-corrector with order {order} (solver type: vary coeff)') ns = self.noise_schedule assert order <= len(model_prev_list) # first compute rks t_prev_0 = t_prev_list[-1] lambda_prev_0 = ns.marginal_lambda(t_prev_0) lambda_t = ns.marginal_lambda(t) model_prev_0 = model_prev_list[-1] sigma_prev_0, sigma_t = ns.marginal_std(t_prev_0), ns.marginal_std(t) log_alpha_t = ns.marginal_log_mean_coeff(t) alpha_t = torch.exp(log_alpha_t) h = lambda_t - lambda_prev_0 rks = [] D1s = [] for i in range(1, order): t_prev_i = t_prev_list[-(i + 1)] model_prev_i = model_prev_list[-(i + 1)] lambda_prev_i = ns.marginal_lambda(t_prev_i) rk = (lambda_prev_i - lambda_prev_0) / h rks.append(rk) D1s.append((model_prev_i - model_prev_0) / rk) rks.append(1.) rks = torch.tensor(rks, device=x.device) K = len(rks) # build C matrix C = [] col = torch.ones_like(rks) for k in range(1, K + 1): C.append(col) col = col * rks / (k + 1) C = torch.stack(C, dim=1) if len(D1s) > 0: D1s = torch.stack(D1s, dim=1) # (B, K) C_inv_p = torch.linalg.inv(C[:-1, :-1]) A_p = C_inv_p if use_corrector: #print('using corrector') C_inv = torch.linalg.inv(C) A_c = C_inv hh = -h if self.predict_x0 else h h_phi_1 = torch.expm1(hh) h_phi_ks = [] factorial_k = 1 h_phi_k = h_phi_1 for k in range(1, K + 2): h_phi_ks.append(h_phi_k) h_phi_k = h_phi_k / hh - 1 / factorial_k factorial_k *= (k + 1) model_t = None if self.predict_x0: x_t_ = ( sigma_t / sigma_prev_0 * x - alpha_t * h_phi_1 * model_prev_0 ) # now predictor x_t = x_t_ if len(D1s) > 0: # compute the residuals for predictor for k in range(K - 1): x_t = x_t - alpha_t * h_phi_ks[k + 1] * torch.einsum('bkchw,k->bchw', D1s, A_p[k]) # now corrector if use_corrector: model_t = self.model_fn(x_t, t) D1_t = (model_t - model_prev_0) x_t = x_t_ k = 0 for k in range(K - 1): x_t = x_t - alpha_t * h_phi_ks[k + 1] * torch.einsum('bkchw,k->bchw', D1s, A_c[k][:-1]) x_t = x_t - alpha_t * h_phi_ks[K] * (D1_t * A_c[k][-1]) else: log_alpha_prev_0, log_alpha_t = ns.marginal_log_mean_coeff(t_prev_0), ns.marginal_log_mean_coeff(t) x_t_ = ( (torch.exp(log_alpha_t - log_alpha_prev_0)) * x - (sigma_t * h_phi_1) * model_prev_0 ) # now predictor x_t = x_t_ if len(D1s) > 0: # compute the residuals for predictor for k in range(K - 1): x_t = x_t - sigma_t * h_phi_ks[k + 1] * torch.einsum('bkchw,k->bchw', D1s, A_p[k]) # now corrector if use_corrector: model_t = self.model_fn(x_t, t) D1_t = (model_t - model_prev_0) x_t = x_t_ k = 0 for k in range(K - 1): x_t = x_t - sigma_t * h_phi_ks[k + 1] * torch.einsum('bkchw,k->bchw', D1s, A_c[k][:-1]) x_t = x_t - sigma_t * h_phi_ks[K] * (D1_t * A_c[k][-1]) return x_t, model_t def multistep_uni_pc_bh_update(self, x, model_prev_list, t_prev_list, t, order, x_t=None, use_corrector=True): #print(f'using unified predictor-corrector with order {order} (solver type: B(h))') ns = self.noise_schedule assert order <= len(model_prev_list) dims = x.dim() # first compute rks t_prev_0 = t_prev_list[-1] lambda_prev_0 = ns.marginal_lambda(t_prev_0) lambda_t = ns.marginal_lambda(t) model_prev_0 = model_prev_list[-1] sigma_prev_0, sigma_t = ns.marginal_std(t_prev_0), ns.marginal_std(t) log_alpha_prev_0, log_alpha_t = ns.marginal_log_mean_coeff(t_prev_0), ns.marginal_log_mean_coeff(t) alpha_t = torch.exp(log_alpha_t) h = lambda_t - lambda_prev_0 rks = [] D1s = [] for i in range(1, order): t_prev_i = t_prev_list[-(i + 1)] model_prev_i = model_prev_list[-(i + 1)] lambda_prev_i = ns.marginal_lambda(t_prev_i) rk = ((lambda_prev_i - lambda_prev_0) / h)[0] rks.append(rk) D1s.append((model_prev_i - model_prev_0) / rk) rks.append(1.) rks = torch.tensor(rks, device=x.device) R = [] b = [] hh = -h[0] if self.predict_x0 else h[0] h_phi_1 = torch.expm1(hh) # h\phi_1(h) = e^h - 1 h_phi_k = h_phi_1 / hh - 1 factorial_i = 1 if self.variant == 'bh1': B_h = hh elif self.variant == 'bh2': B_h = torch.expm1(hh) else: raise NotImplementedError() for i in range(1, order + 1): R.append(torch.pow(rks, i - 1)) b.append(h_phi_k * factorial_i / B_h) factorial_i *= (i + 1) h_phi_k = h_phi_k / hh - 1 / factorial_i R = torch.stack(R) b = torch.tensor(b, device=x.device) # now predictor use_predictor = len(D1s) > 0 and x_t is None if len(D1s) > 0: D1s = torch.stack(D1s, dim=1) # (B, K) if x_t is None: # for order 2, we use a simplified version if order == 2: rhos_p = torch.tensor([0.5], device=b.device) else: rhos_p = torch.linalg.solve(R[:-1, :-1], b[:-1]) else: D1s = None if use_corrector: #print('using corrector') # for order 1, we use a simplified version if order == 1: rhos_c = torch.tensor([0.5], device=b.device) else: rhos_c = torch.linalg.solve(R, b) model_t = None if self.predict_x0: x_t_ = ( expand_dims(sigma_t / sigma_prev_0, dims) * x - expand_dims(alpha_t * h_phi_1, dims)* model_prev_0 ) if x_t is None: if use_predictor: pred_res = torch.einsum('k,bkchw->bchw', rhos_p, D1s) else: pred_res = 0 x_t = x_t_ - expand_dims(alpha_t * B_h, dims) * pred_res if use_corrector: model_t = self.model_fn(x_t, t) if D1s is not None: corr_res = torch.einsum('k,bkchw->bchw', rhos_c[:-1], D1s) else: corr_res = 0 D1_t = (model_t - model_prev_0) x_t = x_t_ - expand_dims(alpha_t * B_h, dims) * (corr_res + rhos_c[-1] * D1_t) else: x_t_ = ( expand_dims(torch.exp(log_alpha_t - log_alpha_prev_0), dims) * x - expand_dims(sigma_t * h_phi_1, dims) * model_prev_0 ) if x_t is None: if use_predictor: pred_res = torch.einsum('k,bkchw->bchw', rhos_p, D1s) else: pred_res = 0 x_t = x_t_ - expand_dims(sigma_t * B_h, dims) * pred_res if use_corrector: model_t = self.model_fn(x_t, t) if D1s is not None: corr_res = torch.einsum('k,bkchw->bchw', rhos_c[:-1], D1s) else: corr_res = 0 D1_t = (model_t - model_prev_0) x_t = x_t_ - expand_dims(sigma_t * B_h, dims) * (corr_res + rhos_c[-1] * D1_t) return x_t, model_t 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, corrector=False, ): 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 == 'multistep': assert steps >= order, "UniPC order must be < sampling steps" timesteps = self.get_time_steps(skip_type=skip_type, t_T=t_T, t_0=t_0, N=steps, device=device) #print(f"Running UniPC Sampling with {timesteps.shape[0]} timesteps, order {order}") 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] with tqdm.tqdm(total=steps) as pbar: # 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, model_x = self.multistep_uni_pc_update(x, model_prev_list, t_prev_list, vec_t, init_order, use_corrector=True) if model_x is None: model_x = self.model_fn(x, vec_t) if self.after_update is not None: self.after_update(x, model_x) model_prev_list.append(model_x) t_prev_list.append(vec_t) pbar.update() for step in range(order, steps + 1): vec_t = timesteps[step].expand(x.shape[0]) if lower_order_final: step_order = min(order, steps + 1 - step) else: step_order = order #print('this step order:', step_order) if step == steps: #print('do not run corrector at the last step') use_corrector = False else: use_corrector = True x, model_x = self.multistep_uni_pc_update(x, model_prev_list, t_prev_list, vec_t, step_order, use_corrector=use_corrector) if self.after_update is not None: self.after_update(x, model_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] = vec_t # We do not need to evaluate the final model value. if step < steps: if model_x is None: model_x = self.model_fn(x, vec_t) model_prev_list[-1] = model_x pbar.update() else: raise NotImplementedError() 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)]