SongDiffusionSDE

class deepinv.sampling.SongDiffusionSDE(beta_t=None, B_t=None, xi_t=None, variance_preserving=False, variance_exploding=False, alpha=0.0, T=1.0, denoiser=None, solver=None, dtype=torch.float64, device=torch.device('cpu'), *args, **kwargs)

Bases: EDMDiffusionSDE

Generative diffusion Stochastic Differential Equation.

This class implements the diffusion generative SDE based the formulation from Song et al.[1]:

\[d x_t = -\left(\frac{1}{2} \beta(t) x_t + \frac{1 + \alpha(t)}{2} \xi(t) \nabla \log p_t(x_t) \right) dt + \sqrt{\alpha(t) \xi(t)} d w_t\]

where \(\beta(t)\) is a time-dependent linear drift, \(\xi(t)\) is a time-dependent linear diffusion, and \(\alpha(t)\) is weighting the diffusion term.

It corresponds to the reverse-time SDE of the following forward-time SDE:

\[d x_t = -\frac{1}{2} \beta(t) x_t dt + \sqrt{\xi(t)} d w_t\]

Compared to the EDM formulation in deepinv.sampling.EDMDiffusionSDE, the scale \(s(t)\) and noise \(\sigma(t)\) schedulers are defined with respect to \(\beta(t)\) and \(\xi(t)\) as follows:

\[s(t) = \exp\left(-\int_0^t \beta(s) ds\right), \quad \sigma(t) = \sqrt{2 \int_0^t \frac{\xi(s)}{s(s)^2} ds}.\]

Common choices include the variance-preserving formulation \(\beta(t) = \xi(t)\) and the variance-exploding formulation \(\beta(t) = 0\).

• For choosing variance-preserving formulation, set variance_preserving=True and beta_t and xi_t will be automatically set to be the same function.

• For choosing variance-exploding formulation, set variance_exploding=True and beta_t will be automatically set to 0.

Note

This SDE must be solved going reverse in time i.e. from \(t=T\) to \(t=0\).

Parameters:

• beta_t (Callable) – a time-dependent linear drift of the forward-time SDE.

• B_t (Callable) – time integral of beta_t between 0 and t. If None, it is calculated by numerical integration.

• xi_t (Callable) – a time-dependent linear diffusion of the forward-time SDE.

• denoiser (deepinv.models.Denoiser) – a denoiser used to provide an approximation of the score at time \(t\): \(\nabla \log p_t\).

• alpha (Callable, float) – a (possibly time-dependent) positive scalar weighting the diffusion term. A constant function \(\alpha(t) = 0\) corresponds to ODE sampling and \(\alpha(t) > 0\) corresponds to SDE sampling.

• T (float) – the end time of the forward SDE.

• solver (deepinv.sampling.BaseSDESolver) – the solver for solving the SDE.

• dtype (torch.dtype) – data type of the computation, except for the denoiser which will use torch.float32. We recommend using torch.float64 for better stability and less numerical error when solving the SDE in discrete time, since most computation cost is from evaluating the denoiser, which will be always computed in torch.float32.

• device (torch.device) – device on which the computation is performed.

• *args – additional arguments for the deepinv.sampling.DiffusionSDE.

• **kwargs – additional keyword arguments for the deepinv.sampling.DiffusionSDE.

---

References:

[1]

Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. In International Conference on Learning Representations. 2020.

Examples using SongDiffusionSDE:

Building your diffusion posterior sampling method using SDEs

Building your diffusion posterior sampling method using SDEs