VarianceExplodingDiffusion

class deepinv.sampling.VarianceExplodingDiffusion(denoiser=None, sigma_min=0.005, sigma_max=100, alpha=1.0, solver=None, dtype=torch.float64, device=torch.device('cpu'), *args, **kwargs)

Bases: DiffusionSDE

Variance-Exploding Stochastic Differential Equation (VE-SDE).

This class implements the reverse-time SDE of the Variance-Exploding SDE (VE-SDE) Song et al.[1].

The forward-time SDE is defined as follows:

\[d x_t = g(t) d w_t \quad \mbox{where } g(t) = \sigma_{\mathrm{min}} \left( \frac{\sigma_{\mathrm{max}}}{\sigma_{\mathrm{min}}} \right)^t\]

The reverse-time SDE is defined as follows:

\[d x_t = -\frac{1 + \alpha}{2} \sigma_{\mathrm{min}}^2 \left( \frac{\sigma_{\mathrm{max}}}{\sigma_{\mathrm{min}}} \right)^{2t} \nabla \log p_t(x_t) dt + \sqrt{\alpha} \sigma_{\mathrm{min}} \left( \frac{\sigma_{\mathrm{max}}}{\sigma_{\mathrm{min}}} \right)^t d w_t\]

where \(\alpha \in [0,1]\) is a constant weighting the diffusion term.

This class is the reverse-time SDE of the VE-SDE, serving as the generation process.

Parameters:

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

• sigma_min (float) – the minimum noise level.

• sigma_max (float) – the maximum noise level.

• alpha (float) – the weighting factor of the diffusion term.

• 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.

sample_init(shape, rng)

Sample from the initial distribution of the reverse-time diffusion SDE, which is a Gaussian with zero mean and covariance matrix \(\sigma_{max}^2 \operatorname{Id}\).

Parameters:

• shape (tuple) – The shape of the sample to generate

• rng (torch.Generator) – Random number generator for reproducibility

Returns:

A sample from the prior distribution

Return type:

torch.Tensor

Examples using VarianceExplodingDiffusion:

Building your diffusion posterior sampling method using SDEs

Building your diffusion posterior sampling method using SDEs