定义SDE类
定义了7个子函数
T: End time of the SDE.
sde:
marginal_prob: Parameters to determine the marginal distribution of the SDE,
p
t
(
x
)
p_t(x)
pt(x).
prior_sampling: Generate one sample from the prior distribution,
p
T
(
x
)
p_T(x)
pT(x).
prior_logp: Compute log-density of the prior distribution.
discretize: Discretize the SDE in the form: x_{i+1} = x_i + f_i(x_i) + G_i z_i.
Useful for reverse diffusion sampling and probabiliy flow sampling.
Defaults to Euler-Maruyama discretization.
reverse: Create the reverse-time SDE/ODE.
reverse
class SDE(abc.ABC):
"""SDE abstract class. Functions are designed for a mini-batch of inputs."""
def __init__(self, N):
"""Construct an SDE.
Args:
N: number of discretization time steps.
"""
super().__init__()
self.N = N
@property
@abc.abstractmethod
def T(self):
"""End time of the SDE."""
pass
@abc.abstractmethod
def sde(self, x, t):
pass
@abc.abstractmethod
def marginal_prob(self, x, t):
"""Parameters to determine the marginal distribution of the SDE, $p_t(x)$."""
pass
@abc.abstractmethod
def prior_sampling(self, shape):
"""Generate one sample from the prior distribution, $p_T(x)$."""
pass
@abc.abstractmethod
def prior_logp(self, z):
"""Compute log-density of the prior distribution.
Useful for computing the log-likelihood via probability flow ODE.
Args:
z: latent code
Returns:
log probability density
"""
pass
def discretize(self, x, t):
"""Discretize the SDE in the form: x_{i+1} = x_i + f_i(x_i) + G_i z_i.
Useful for reverse diffusion sampling and probabiliy flow sampling.
Defaults to Euler-Maruyama discretization.
Args:
x: a torch tensor
t: a torch float representing the time step (from 0 to `self.T`)
Returns:
f, G
"""
dt = 1 / self.N
drift, diffusion = self.sde(x, t)
f = drift * dt
G = diffusion * torch.sqrt(torch.tensor(dt, device=t.device))
return f, G
def reverse(self, score_fn, probability_flow=False):
"""Create the reverse-time SDE/ODE.
Args:
score_fn: A time-dependent score-based model that takes x and t and returns the score.
probability_flow: If `True`, create the reverse-time ODE used for probability flow sampling.
"""
N = self.N
T = self.T
sde_fn = self.sde
discretize_fn = self.discretize
定义reverse-time SDE类
# Build the class for reverse-time SDE.
class RSDE(self.__class__):
def __init__(self):
self.N = N
self.probability_flow = probability_flow
@property
def T(self):
return T
def sde(self, x, t):
"""Create the drift and diffusion functions for the reverse SDE/ODE."""
drift, diffusion = sde_fn(x, t)
score = score_fn(x, t)
drift = drift - diffusion[:, None, None, None] ** 2 * score * (0.5 if self.probability_flow else 1.)
# Set the diffusion function to zero for ODEs.
diffusion = 0. if self.probability_flow else diffusion
return drift, diffusion
def discretize(self, x, t):
"""Create discretized iteration rules for the reverse diffusion sampler."""
f, G = discretize_fn(x, t)
rev_f = f - G[:, None, None, None] ** 2 * score_fn(x, t) * (0.5 if self.probability_flow else 1.)
rev_G = torch.zeros_like(G) if self.probability_flow else G
return rev_f, rev_G
return RSDE()
定义VPSDE类
class VPSDE(SDE):
def __init__(self, beta_min=0.1, beta_max=20, N=1000):
"""Construct a Variance Preserving SDE.
Args:
beta_min: value of beta(0)
beta_max: value of beta(1)
N: number of discretization steps
"""
super().__init__(N)
self.beta_0 = beta_min
self.beta_1 = beta_max
self.N = N
self.discrete_betas = torch.linspace(beta_min / N, beta_max / N, N)
self.alphas = 1. - self.discrete_betas
self.alphas_cumprod = torch.cumprod(self.alphas, dim=0)
self.sqrt_alphas_cumprod = torch.sqrt(self.alphas_cumprod)
self.sqrt_1m_alphas_cumprod = torch.sqrt(1. - self.alphas_cumprod)
@property
def T(self):
return 1
def sde(self, x, t):
beta_t = self.beta_0 + t * (self.beta_1 - self.beta_0)
drift = -0.5 * beta_t[:, None, None, None] * x
diffusion = torch.sqrt(beta_t)
return drift, diffusion
def marginal_prob(self, x, t):
log_mean_coeff = -0.25 * t ** 2 * (self.beta_1 - self.beta_0) - 0.5 * t * self.beta_0
mean = torch.exp(log_mean_coeff[:, None, None, None]) * x
std = torch.sqrt(1. - torch.exp(2. * log_mean_coeff))
return mean, std
def prior_sampling(self, shape):
return torch.randn(*shape)
def prior_logp(self, z):
shape = z.shape
N = np.prod(shape[1:])
logps = -N / 2. * np.log(2 * np.pi) - torch.sum(z ** 2, dim=(1, 2, 3)) / 2.
return logps
def discretize(self, x, t):
"""DDPM discretization."""
timestep = (t * (self.N - 1) / self.T).long()
beta = self.discrete_betas.to(x.device)[timestep]
alpha = self.alphas.to(x.device)[timestep]
sqrt_beta = torch.sqrt(beta)
f = torch.sqrt(alpha)[:, None, None, None] * x - x
G = sqrt_beta
return f, G
定义subVPSDE类
class subVPSDE(SDE):
def __init__(self, beta_min=0.1, beta_max=20, N=1000):
"""Construct the sub-VP SDE that excels at likelihoods.
Args:
beta_min: value of beta(0)
beta_max: value of beta(1)
N: number of discretization steps
"""
super().__init__(N)
self.beta_0 = beta_min
self.beta_1 = beta_max
self.N = N
@property
def T(self):
return 1
def sde(self, x, t):
beta_t = self.beta_0 + t * (self.beta_1 - self.beta_0)
drift = -0.5 * beta_t[:, None, None, None] * x
discount = 1. - torch.exp(-2 * self.beta_0 * t - (self.beta_1 - self.beta_0) * t ** 2)
diffusion = torch.sqrt(beta_t * discount)
return drift, diffusion
#边际概率函数，返回值是边际概率的mean和std
def marginal_prob(self, x, t):
log_mean_coeff = -0.25 * t ** 2 * (self.beta_1 - self.beta_0) - 0.5 * t * self.beta_0
mean = torch.exp(log_mean_coeff)[:, None, None, None] * x
std = 1 - torch.exp(2. * log_mean_coeff)
return mean, std
def prior_sampling(self, shape):
return torch.randn(*shape)
def prior_logp(self, z):
shape = z.shape
N = np.prod(shape[1:])
return -N / 2. * np.log(2 * np.pi) - torch.sum(z ** 2, dim=(1, 2, 3)) / 2.
定义VESDE类
class VESDE(SDE):
def __init__(self, sigma_min=0.01, sigma_max=50, N=1000):
"""Construct a Variance Exploding SDE.
Args:
sigma_min: smallest sigma.
sigma_max: largest sigma.
N: number of discretization steps
"""
super().__init__(N)
self.sigma_min = sigma_min
self.sigma_max = sigma_max
self.discrete_sigmas = torch.exp(torch.linspace(np.log(self.sigma_min), np.log(self.sigma_max), N))
self.N = N
@property
def T(self):
return 1
def sde(self, x, t):
sigma = self.sigma_min * (self.sigma_max / self.sigma_min) ** t
drift = torch.zeros_like(x)
diffusion = sigma * torch.sqrt(torch.tensor(2 * (np.log(self.sigma_max) - np.log(self.sigma_min)),
device=t.device))
return drift, diffusion
def marginal_prob(self, x, t):
std = self.sigma_min * (self.sigma_max / self.sigma_min) ** t
mean = x
return mean, std
def prior_sampling(self, shape):
return torch.randn(*shape) * self.sigma_max
def prior_logp(self, z):
shape = z.shape
N = np.prod(shape[1:])
return -N / 2. * np.log(2 * np.pi * self.sigma_max ** 2) - torch.sum(z ** 2, dim=(1, 2, 3)) / (2 * self.sigma_max ** 2)
def discretize(self, x, t):
"""SMLD(NCSN) discretization."""
timestep = (t * (self.N - 1) / self.T).long()
sigma = self.discrete_sigmas.to(t.device)[timestep]
adjacent_sigma = torch.where(timestep == 0, torch.zeros_like(t),
self.discrete_sigmas[timestep - 1].to(t.device))
f = torch.zeros_like(x)
G = torch.sqrt(sigma ** 2 - adjacent_sigma ** 2)
return f, G
