File size: 9,537 Bytes

6fecfbe

import torch
from torch import no_grad, FloatTensor
from tqdm import tqdm
from typing import Protocol, Optional, Dict, Any, TypedDict, NamedTuple, Union, List
import math
from itertools import tee

def pairwise(iterable):
    "s -> (s0, s1), (s1, s2), (s2, s3), ..."
    a, b = tee(iterable)
    next(b, None)
    return zip(a, b)

class DenoiserModel(Protocol):
  def __call__(self, x: FloatTensor, t: FloatTensor, *args, **kwargs) -> FloatTensor: ...

class RefinedExpCallbackPayload(TypedDict):
  x: FloatTensor
  i: int
  sigma: FloatTensor
  sigma_hat: FloatTensor

class RefinedExpCallback(Protocol):
  def __call__(self, payload: RefinedExpCallbackPayload) -> None: ...

class NoiseSampler(Protocol):
  def __call__(self, x: FloatTensor) -> FloatTensor: ...

class StepOutput(NamedTuple):
  x_next: FloatTensor
  denoised: FloatTensor
  denoised2: FloatTensor

def _gamma(
  n: int,
) -> int:
  """
  https://en.wikipedia.org/wiki/Gamma_function
  for every positive integer n,
  Γ(n) = (n-1)!
  """
  return math.factorial(n-1)

def _incomplete_gamma(
  s: int,
  x: float,
  gamma_s: Optional[int] = None
) -> float:
  """
  https://en.wikipedia.org/wiki/Incomplete_gamma_function#Special_values
  if s is a positive integer,
  Γ(s, x) = (s-1)!*∑{k=0..s-1}(x^k/k!)
  """
  if gamma_s is None:
    gamma_s = _gamma(s)

  sum_: float = 0
  # {k=0..s-1} inclusive
  for k in range(s):
    numerator: float = x**k
    denom: int = math.factorial(k)
    quotient: float = numerator/denom
    sum_ += quotient
  incomplete_gamma_: float = sum_ * math.exp(-x) * gamma_s
  return incomplete_gamma_

# by Katherine Crowson
def _phi_1(neg_h: FloatTensor):
  return torch.nan_to_num(torch.expm1(neg_h) / neg_h, nan=1.0)

# by Katherine Crowson
def _phi_2(neg_h: FloatTensor):
  return torch.nan_to_num((torch.expm1(neg_h) - neg_h) / neg_h**2, nan=0.5)

# by Katherine Crowson
def _phi_3(neg_h: FloatTensor):
  return torch.nan_to_num((torch.expm1(neg_h) - neg_h - neg_h**2 / 2) / neg_h**3, nan=1 / 6)

def _phi(
  neg_h: float,
  j: int,
):
  """
  For j={1,2,3}: you could alternatively use Kat's phi_1, phi_2, phi_3 which perform fewer steps

  Lemma 1
  https://arxiv.org/abs/2308.02157
  ϕj(-h) = 1/h^j*∫{0..h}(e^(τ-h)*(τ^(j-1))/((j-1)!)dτ)

  https://www.wolframalpha.com/input?i=integrate+e%5E%28%CF%84-h%29*%28%CF%84%5E%28j-1%29%2F%28j-1%29%21%29d%CF%84
  = 1/h^j*[(e^(-h)*(-τ)^(-j)*τ(j))/((j-1)!)]{0..h}
  https://www.wolframalpha.com/input?i=integrate+e%5E%28%CF%84-h%29*%28%CF%84%5E%28j-1%29%2F%28j-1%29%21%29d%CF%84+between+0+and+h
  = 1/h^j*((e^(-h)*(-h)^(-j)*h^j*(Γ(j)-Γ(j,-h)))/(j-1)!)
  = (e^(-h)*(-h)^(-j)*h^j*(Γ(j)-Γ(j,-h))/((j-1)!*h^j)
  = (e^(-h)*(-h)^(-j)*(Γ(j)-Γ(j,-h))/(j-1)!
  = (e^(-h)*(-h)^(-j)*(Γ(j)-Γ(j,-h))/Γ(j)
  = (e^(-h)*(-h)^(-j)*(1-Γ(j,-h)/Γ(j))

  requires j>0
  """
  assert j > 0
  gamma_: float = _gamma(j)
  incomp_gamma_: float = _incomplete_gamma(j, neg_h, gamma_s=gamma_)

  phi_: float = math.exp(neg_h) * neg_h**-j * (1-incomp_gamma_/gamma_)

  return phi_

class RESDECoeffsSecondOrder(NamedTuple):
  a2_1: float
  b1: float
  b2: float

def _de_second_order(
  h: float,
  c2: float,
  simple_phi_calc = False,
) -> RESDECoeffsSecondOrder:
  """
  Table 3
  https://arxiv.org/abs/2308.02157
  ϕi,j := ϕi,j(-h) = ϕi(-cj*h)
  a2_1 = c2ϕ1,2
       = c2ϕ1(-c2*h)
  b1 = ϕ1 - ϕ2/c2
  """
  if simple_phi_calc:
    # Kat computed simpler expressions for phi for cases j={1,2,3}
    a2_1: float = c2 * _phi_1(-c2*h)
    phi1: float = _phi_1(-h)
    phi2: float = _phi_2(-h)
  else:
    # I computed general solution instead.
    # they're close, but there are slight differences. not sure which would be more prone to numerical error.
    a2_1: float = c2 * _phi(j=1, neg_h=-c2*h)
    phi1: float = _phi(j=1, neg_h=-h)
    phi2: float = _phi(j=2, neg_h=-h)
  phi2_c2: float = phi2/c2
  b1: float = phi1 - phi2_c2
  b2: float = phi2_c2
  return RESDECoeffsSecondOrder(
    a2_1=a2_1,
    b1=b1,
    b2=b2,
  )  

def _refined_exp_sosu_step(
  model: DenoiserModel,
  x: FloatTensor,
  sigma: FloatTensor,
  sigma_next: FloatTensor,
  c2 = 0.5,
  extra_args: Dict[str, Any] = {},
  pbar: Optional[tqdm] = None,
  simple_phi_calc = False,
) -> StepOutput:
  """
  Algorithm 1 "RES Second order Single Update Step with c2"
  https://arxiv.org/abs/2308.02157

  Parameters:
    model (`DenoiserModel`): a k-diffusion wrapped denoiser model (e.g. a subclass of DiscreteEpsDDPMDenoiser)
    x (`FloatTensor`): noised latents (or RGB I suppose), e.g. torch.randn((B, C, H, W)) * sigma[0]
    sigma (`FloatTensor`): timestep to denoise
    sigma_next (`FloatTensor`): timestep+1 to denoise
    c2 (`float`, *optional*, defaults to .5): partial step size for solving ODE. .5 = midpoint method
    extra_args (`Dict[str, Any]`, *optional*, defaults to `{}`): kwargs to pass to `model#__call__()`
    pbar (`tqdm`, *optional*, defaults to `None`): progress bar to update after each model call
    simple_phi_calc (`bool`, *optional*, defaults to `True`): True = calculate phi_i,j(-h) via simplified formulae specific to j={1,2}. False = Use general solution that works for any j. Mathematically equivalent, but could be numeric differences.
  """
  lam_next, lam = (s.log().neg() for s in (sigma_next, sigma))

  # type hints aren't strictly true regarding float vs FloatTensor.
  # everything gets promoted to `FloatTensor` after interacting with `sigma: FloatTensor`.
  # I will use float to indicate any variables which are scalars.
  h: float = lam_next - lam
  a2_1, b1, b2 = _de_second_order(h=h, c2=c2, simple_phi_calc=simple_phi_calc)
  
  denoised: FloatTensor = model(x, sigma, **extra_args)
  if pbar is not None:
    pbar.update(0.5)

  c2_h: float = c2*h

  x_2: FloatTensor = math.exp(-c2_h)*x + a2_1*h*denoised
  lam_2: float = lam + c2_h
  sigma_2: float = lam_2.neg().exp()

  denoised2: FloatTensor = model(x_2, sigma_2, **extra_args)
  if pbar is not None:
    pbar.update(0.5)

  x_next: FloatTensor = math.exp(-h)*x + h*(b1*denoised + b2*denoised2)
  
  return StepOutput(
    x_next=x_next,
    denoised=denoised,
    denoised2=denoised2,
  )
  

@no_grad()
def sample_refined_exp_s(
  model: FloatTensor,
  x: FloatTensor,
  sigmas: FloatTensor,
  denoise_to_zero: bool = True,
  extra_args: Dict[str, Any] = {},
  callback: Optional[RefinedExpCallback] = None,
  disable: Optional[bool] = None,
  ita: FloatTensor = torch.zeros((1,)),
  c2 = .5,
  noise_sampler: NoiseSampler = torch.randn_like,
  simple_phi_calc = True,
):
  """
  Refined Exponential Solver (S).
  Algorithm 2 "RES Single-Step Sampler" with Algorithm 1 second-order step
  https://arxiv.org/abs/2308.02157

  Parameters:
    model (`DenoiserModel`): a k-diffusion wrapped denoiser model (e.g. a subclass of DiscreteEpsDDPMDenoiser)
    x (`FloatTensor`): noised latents (or RGB I suppose), e.g. torch.randn((B, C, H, W)) * sigma[0]
    sigmas (`FloatTensor`): sigmas (ideally an exponential schedule!) e.g. get_sigmas_exponential(n=25, sigma_min=model.sigma_min, sigma_max=model.sigma_max)
    denoise_to_zero (`bool`, *optional*, defaults to `True`): whether to finish with a first-order step down to 0 (rather than stopping at sigma_min). True = fully denoise image. False = match Algorithm 2 in paper
    extra_args (`Dict[str, Any]`, *optional*, defaults to `{}`): kwargs to pass to `model#__call__()`
    callback (`RefinedExpCallback`, *optional*, defaults to `None`): you can supply this callback to see the intermediate denoising results, e.g. to preview each step of the denoising process
    disable (`bool`, *optional*, defaults to `False`): whether to hide `tqdm`'s progress bar animation from being printed
    ita (`FloatTensor`, *optional*, defaults to 0.): degree of stochasticity, η, for each timestep. tensor shape must be broadcastable to 1-dimensional tensor with length `len(sigmas) if denoise_to_zero else len(sigmas)-1`. each element should be from 0 to 1.
    c2 (`float`, *optional*, defaults to .5): partial step size for solving ODE. .5 = midpoint method
    noise_sampler (`NoiseSampler`, *optional*, defaults to `torch.randn_like`): method used for adding noise
    simple_phi_calc (`bool`, *optional*, defaults to `True`): True = calculate phi_i,j(-h) via simplified formulae specific to j={1,2}. False = Use general solution that works for any j. Mathematically equivalent, but could be numeric differences.
  """
  # assert sigmas[-1] == 0
  ita = ita.to(x.device)
  with tqdm(disable=disable, total=len(sigmas)-(1 if denoise_to_zero else 2)) as pbar:
    for i, (sigma, sigma_next) in enumerate(pairwise(sigmas[:-1].split(1))):
      eps: FloatTensor = noise_sampler(x)
      sigma_hat = sigma * (1 + ita)
      x_hat = x + (sigma_hat ** 2 - sigma ** 2) ** .5 * eps
      x_next, denoised, denoised2 = _refined_exp_sosu_step(
        model,
        x_hat,
        sigma_hat,
        sigma_next,
        c2=c2,
        extra_args=extra_args,
        pbar=pbar,
        simple_phi_calc=simple_phi_calc,
      )
      if callback is not None:
        payload = RefinedExpCallbackPayload(
          x=x,
          i=i,
          sigma=sigma,
          sigma_hat=sigma_hat,
          denoised=denoised,
          denoised2=denoised2,
        )
        callback(payload)
      x = x_next
    if denoise_to_zero:
      eps: FloatTensor = noise_sampler(x)
      sigma_hat = sigma * (1 + ita)
      x_hat = x + (sigma_hat ** 2 - sigma ** 2) ** .5 * eps
      x_next: FloatTensor = model(x_hat, sigma.to(x_hat.device))
      pbar.update()
      x = x_next
  return x