class

VESDE

extendsSDE

VESDE(sigma_min: float = 0.01, sigma_max: float = 50.0)

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Variance Exploding — the continuous limit of NCSN.

Parameters

sigma_minfloat= 0.01

The geometric noise range.

sigma_maxfloat= 0.01

The geometric noise range.

Notes

Song et al. (2021), equation 30:

dx = \sigma_{\min}\left(\frac{\sigma_{\max}}{\sigma_{\min}}\right)^t \sqrt{2\log\frac{\sigma_{\max}}{\sigma_{\min}}}\; dw, \qquad t \in (0, 1],

with the perturbation kernel of equation 31, $\mathcal{N}(x(t); x(0), \sigma(t)^2 I)$ — the mean does not move, which is what "variance exploding" means.

Defined on the open interval. The paper is explicit that $\sigma$ is not differentiable at zero because $\sigma(0) = 0$ by the discrete convention while $\sigma(0^+) = \sigma_{\min}$ , "causing the VE SDE ... undefined for t = 0". Sampling therefore stops at t_min rather than running to zero.

Examples

>>> from lucid.models.generative.score_sde._sde import VESDE
>>> sde = VESDE(sigma_min=0.01, sigma_max=50.0)
>>> sde.prior_variance
2500.0
Variance exploding: the prior is as wide as sigma_max squared, so
the terminal distribution is nothing like the data and the model
learns the whole way down.

Used by 2

Constructors

dunder

init

→None

__init__(sigma_min: float = 0.01, sigma_max: float = 50.0)

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Initialise the SDE. See the class docstring for parameters.

Properties

prop

prior_variance

→float

prior_variance: float

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$\sigma_{\max}^2$ .

prop

t_min

→float

t_min: float

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Sampling stops here — see the class docstring on t = 0.

Instance methods

diffusion

→Tensor

diffusion(t: Tensor)

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Equation 30's coefficient.

drift

→Tensor

drift(x: Tensor, t: Tensor)

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Zero — the VE process only adds variance.

marginal_prob

→tuple[Tensor, Tensor]

marginal_prob(x: Tensor, t: Tensor)

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Equation 31 — the mean is the sample itself.

prior_sampling

→Tensor

prior_sampling(shape: tuple[int, ...], device: str)

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$\mathcal{N}(0, \sigma_{\max}^2 I)$ .

sigma

→Tensor

sigma(t: Tensor)

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$\sigma(t) = \sigma_{\min}(\sigma_{\max}/\sigma_{\min})^t$ .

>>> from lucid.models.generative.score_sde._sde import VESDE >>> sde = VESDE(sigma_min=0.01, sigma_max=50.0) >>> sde.prior_variance 2500.0 Variance exploding: the prior is as wide as sigma_max squared, so the terminal distribution is nothing like the data and the model learns the whole way down.