File size: 65,969 Bytes
447ff7c |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 |
import torch
import torch.nn.functional as F
import math
from tqdm 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(
"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 tqdm(range(1, order), desc="DPM init 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 tqdm(range(order, steps + 1), desc="DPM multistep"):
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)] |