332 lines
12 KiB
Python
332 lines
12 KiB
Python
"""Abstract SDE classes, Reverse SDE, and VP SDEs."""
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import abc
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import torch
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import numpy as np
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class SDE(abc.ABC):
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"""SDE abstract class. Functions are designed for a mini-batch of inputs."""
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def __init__(self, N):
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"""Construct an SDE.
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Args:
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N: number of discretization time steps.
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"""
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super().__init__()
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self.N = N
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@property
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@abc.abstractmethod
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def T(self):
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"""End time of the SDE."""
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pass
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@abc.abstractmethod
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def sde(self, x, t):
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pass
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@abc.abstractmethod
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def marginal_prob(self, x, t):
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"""Parameters to determine the marginal distribution of the SDE, $p_t(x)$"""
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pass
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@abc.abstractmethod
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def prior_sampling(self, shape):
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"""Generate one sample from the prior distribution, $p_T(x)$."""
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pass
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@abc.abstractmethod
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def prior_logp(self, z, mask):
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"""Compute log-density of the prior distribution.
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Useful for computing the log-likelihood via probability flow ODE.
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Args:
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z: latent code
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Returns:
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log probability density
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"""
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pass
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def discretize(self, x, t):
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"""Discretize the SDE in the form: x_{i+1} = x_i + f_i(x_i) + G_i z_i.
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Useful for reverse diffusion sampling and probability flow sampling.
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Defaults to Euler-Maruyama discretization.
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Args:
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x: a torch tensor
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t: a torch float representing the time step (from 0 to `self.T`)
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Returns:
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f, G
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"""
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dt = 1 / self.N
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drift, diffusion = self.sde(x, t)
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f = drift * dt
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G = diffusion * torch.sqrt(torch.tensor(dt, device=t.device))
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return f, G
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def reverse(self, score_fn, probability_flow=False):
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"""Create the reverse-time SDE/ODE.
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Args:
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score_fn: A time-dependent score-based model that takes x and t and returns the score.
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probability_flow: If `True`, create the reverse-time ODE used for probability flow sampling.
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"""
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N = self.N
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T = self.T
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sde_fn = self.sde
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discretize_fn = self.discretize
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# Build the class for reverse-time SDE.
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class RSDE(self.__class__):
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def __init__(self):
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self.N = N
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self.probability_flow = probability_flow
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@property
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def T(self):
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return T
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def sde(self, x, t, *args, **kwargs):
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"""Create the drift and diffusion functions for the reverse SDE/ODE."""
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drift, diffusion = sde_fn(x, t)
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score = score_fn(x, t, *args, **kwargs)
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drift = drift - diffusion[:, None, None] ** 2 * score * (0.5 if self.probability_flow else 1.)
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# Set the diffusion function to zero for ODEs.
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diffusion = 0. if self.probability_flow else diffusion
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return drift, diffusion
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'''
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def sde_score(self, x, t, score):
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"""Create the drift and diffusion functions for the reverse SDE/ODE, given score values."""
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drift, diffusion = sde_fn(x, t)
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if len(score.shape) == 4:
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drift = drift - diffusion[:, None, None, None] ** 2 * score * (0.5 if self.probability_flow else 1.)
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elif len(score.shape) == 3:
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drift = drift - diffusion[:, None, None] ** 2 * score * (0.5 if self.probability_flow else 1.)
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else:
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raise ValueError
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diffusion = 0. if self.probability_flow else diffusion
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return drift, diffusion
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'''
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def discretize(self, x, t, *args, **kwargs):
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"""Create discretized iteration rules for the reverse diffusion sampler."""
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f, G = discretize_fn(x, t)
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rev_f = f - G[:, None, None] ** 2 * score_fn(x, t, *args, **kwargs) * \
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(0.5 if self.probability_flow else 1.)
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rev_G = torch.zeros_like(G) if self.probability_flow else G
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return rev_f, rev_G
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'''
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def discretize_score(self, x, t, score):
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"""Create discretized iteration rules for the reverse diffusion sampler, given score values."""
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f, G = discretize_fn(x, t)
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if len(score.shape) == 4:
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rev_f = f - G[:, None, None, None] ** 2 * score * \
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(0.5 if self.probability_flow else 1.)
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elif len(score.shape) == 3:
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rev_f = f - G[:, None, None] ** 2 * score * (0.5 if self.probability_flow else 1.)
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else:
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raise ValueError
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rev_G = torch.zeros_like(G) if self.probability_flow else G
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return rev_f, rev_G
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'''
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return RSDE()
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class VPSDE(SDE):
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def __init__(self, beta_min=0.1, beta_max=20, N=1000):
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"""Construct a Variance Preserving SDE.
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Args:
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beta_min: value of beta(0)
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beta_max: value of beta(1)
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N: number of discretization steps
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"""
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super().__init__(N)
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self.beta_0 = beta_min
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self.beta_1 = beta_max
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self.N = N
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self.discrete_betas = torch.linspace(beta_min / N, beta_max / N, N)
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self.alphas = 1. - self.discrete_betas
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self.alphas_cumprod = torch.cumprod(self.alphas, dim=0)
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self.sqrt_alphas_cumprod = torch.sqrt(self.alphas_cumprod)
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self.sqrt_1m_alphas_cumprod = torch.sqrt(1. - self.alphas_cumprod)
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@property
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def T(self):
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return 1
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def sde(self, x, t):
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beta_t = self.beta_0 + t * (self.beta_1 - self.beta_0)
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if len(x.shape) == 4:
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drift = -0.5 * beta_t[:, None, None, None] * x
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elif len(x.shape) == 3:
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drift = -0.5 * beta_t[:, None, None] * x
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else:
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raise NotImplementedError
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diffusion = torch.sqrt(beta_t)
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return drift, diffusion
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def marginal_prob(self, x, t):
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log_mean_coeff = -0.25 * t ** 2 * (self.beta_1 - self.beta_0) - 0.5 * t * self.beta_0
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if len(x.shape) == 4:
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mean = torch.exp(log_mean_coeff[:, None, None, None]) * x
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elif len(x.shape) == 3:
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mean = torch.exp(log_mean_coeff[:, None, None]) * x
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else:
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raise ValueError("The shape of x in marginal_prob is not correct.")
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std = torch.sqrt(1. - torch.exp(2. * log_mean_coeff))
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return mean, std
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# def log_snr(self, t):
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# log_mean_coeff = -0.25 * t ** 2 * (self.beta_1 - self.beta_0) - 0.5 * t * self.beta_0
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# mean = torch.exp(log_mean_coeff)
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# std = torch.sqrt(1. - torch.exp(2. * log_mean_coeff))
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# log_snr = torch.log(mean / std)
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# return log_snr, mean, std
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#
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# def log_snr_np(self, t):
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# log_mean_coeff = -0.25 * t ** 2 * (self.beta_1 - self.beta_0) - 0.5 * t * self.beta_0
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# mean = np.exp(log_mean_coeff)
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# std = np.sqrt(1. - np.exp(2. * log_mean_coeff))
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# log_snr = np.log(mean / std)
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# return log_snr
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#
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# def lambda2t(self, lambda_ori):
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# log_val = torch.log(torch.exp(-2. * lambda_ori) + 1.)
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# t = 2. * log_val / (torch.sqrt(self.beta_0 ** 2 + 2. * (self.beta_1 - self.beta_0) * log_val) + self.beta_0)
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# return t
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#
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# def lambda2t_np(self, lambda_ori):
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# log_val = np.log(np.exp(-2. * lambda_ori) + 1.)
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# t = 2. * log_val / (np.sqrt(self.beta_0 ** 2 + 2. * (self.beta_1 - self.beta_0) * log_val) + self.beta_0)
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# return t
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# def prior_sampling(self, shape):
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# sample = torch.randn(*shape)
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# if len(shape) == 4:
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# sample = torch.tril(sample, -1)
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# sample = sample + sample.transpose(-1, -2)
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# return sample
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def prior_sampling(self, shape):
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return torch.randn(*shape)
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def prior_logp(self, z, mask):
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N = torch.sum(mask, dim=tuple(range(1, len(mask.shape))))
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logps = -N / 2. * np.log(2 * np.pi) - torch.sum((z * mask) ** 2, dim=(1, 2, 3)) / 2.
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return logps
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def discretize(self, x, t):
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"""DDPM discretization."""
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timestep = (t * (self.N - 1) / self.T).long()
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beta = self.discrete_betas.to(x.device)[timestep]
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alpha = self.alphas.to(x.device)[timestep]
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sqrt_beta = torch.sqrt(beta)
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if len(x.shape) == 4:
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f = torch.sqrt(alpha)[:, None, None, None] * x - x
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elif len(x.shape) == 3:
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f = torch.sqrt(alpha)[:, None, None] * x - x
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else:
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NotImplementedError
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G = sqrt_beta
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return f, G
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class subVPSDE(SDE):
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def __init__(self, beta_min=0.1, beta_max=20, N=1000):
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"""Construct the sub-VP SDE that excels at likelihoods.
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Args:
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beta_min: value of beta(0)
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beta_max: value of beta(1)
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N: number of discretization steps
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"""
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super().__init__(N)
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self.beta_0 = beta_min
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self.beta_1 = beta_max
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self.N = N
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@property
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def T(self):
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return 1
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def sde(self, x, t):
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beta_t = self.beta_0 + t * (self.beta_1 - self.beta_0)
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drift = -0.5 * beta_t[:, None, None] * x
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discount = 1. - torch.exp(-2 * self.beta_0 * t - (self.beta_1 - self.beta_0) * t ** 2)
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diffusion = torch.sqrt(beta_t * discount)
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return drift, diffusion
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def marginal_prob(self, x, t):
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log_mean_coeff = -0.25 * t ** 2 * (self.beta_1 - self.beta_0) - 0.5 * t * self.beta_0
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mean = torch.exp(log_mean_coeff)[:, None, None] * x
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std = 1 - torch.exp(2. * log_mean_coeff)
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return mean, std
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def prior_sampling(self, shape):
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return torch.randn(*shape)
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def prior_logp(self, z):
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shape = z.shape
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N = np.prod(shape[1:])
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return -N / 2. * np.log(2 * np.pi) - torch.sum(z ** 2, dim=(1, 2, 3)) / 2.
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class VESDE(SDE):
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def __init__(self, sigma_min=0.01, sigma_max=50, N=1000):
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"""Construct a Variance Exploding SDE.
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Args:
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sigma_min: smallest sigma.
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sigma_max: largest sigma.
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N: number of discretization steps
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"""
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super().__init__(N)
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self.sigma_min = sigma_min
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self.sigma_max = sigma_max
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self.discrete_sigmas = torch.exp(torch.linspace(np.log(self.sigma_min), np.log(self.sigma_max), N))
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self.N = N
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@property
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def T(self):
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return 1
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def sde(self, x, t):
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sigma = self.sigma_min * (self.sigma_max / self.sigma_min) ** t
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drift = torch.zeros_like(x)
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diffusion = sigma * torch.sqrt(torch.tensor(2 * (np.log(self.sigma_max) - np.log(self.sigma_min)),
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device=t.device))
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return drift, diffusion
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def marginal_prob(self, x, t):
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std = self.sigma_min * (self.sigma_max / self.sigma_min) ** t
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mean = x
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return mean, std
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def prior_sampling(self, shape):
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return torch.randn(*shape) * self.sigma_max
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def prior_logp(self, z):
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shape = z.shape
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N = np.prod(shape[1:])
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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)
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def discretize(self, x, t):
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"""SMLD(NCSN) discretization."""
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timestep = (t * (self.N - 1) / self.T).long()
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sigma = self.discrete_sigmas.to(t.device)[timestep]
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adjacent_sigma = torch.where(timestep == 0, torch.zeros_like(t),
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self.discrete_sigmas[timestep.cpu() - 1].to(t.device))
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f = torch.zeros_like(x)
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G = torch.sqrt(sigma ** 2 - adjacent_sigma ** 2)
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return f, G |