ULAIterator#

class deepinv.sampling.ULAIterator(algo_params, clip=None)[source]#

Bases: SamplingIterator

Projected Plug-and-Play Unadjusted Langevin Algorithm.

The algorithm runs the following markov chain iteration (Algorithm 2 from https://arxiv.org/abs/2103.04715):

\[x_{k+1} = \Pi_{[a,b]} \left(x_{k} + \eta \nabla \log p(y|A,x_k) + \eta \alpha \nabla \log p(x_{k}) + \sqrt{2\eta}z_{k+1} \right).\]

where \(x_{k}\) is the \(k\) th sample of the Markov chain, \(\log p(y|x)\) is the log-likelihood function, \(\log p(x)\) is the log-prior, \(\eta>0\) is the step size, \(\alpha>0\) controls the amount of regularization, \(\Pi_{[a,b]}(x)\) projects the entries of \(x\) to the interval \([a,b]\) and \(z\sim \mathcal{N}(0,I)\) is a standard Gaussian vector.

  • Projected PnP-ULA assumes that the denoiser is \(L\)-Lipschitz differentiable

  • For convergence, ULA requires that step_size is smaller than \(\frac{1}{L+\|A\|_2^2}\)

Parameters:

  • clip (tuple(int,int)) – Tuple of (min, max) values to clip/project the samples into a bounded range during sampling. Useful for images where pixel values should stay within a specific range (e.g., (0,1) or (0,255)). Default: None

  • algo_params (dict) – Dictionary containing the algorithm parameters (see table below)

Parameter

Type

Description

step_size

float

Step size \(\eta\) (default: 1.0)

alpha

float

Regularization parameter \(\alpha\) (default: 1.0)

sigma

float

Noise level for the score model (default: 0.05)
Returns:

Next state \(X_{t+1}\) in the Markov chain

Return type:

torch.Tensor

forward(X, y, physics, cur_data_fidelity, cur_prior, iteration, *args, **kwargs)[source]#

Performs a single ULA sampling step using the Unadjusted Langevin Algorithm.

Computes the next state in the Markov chain using the formula:

\[x_{t+1} = x_t + \eta \nabla \log p(y|A,x_t) + \eta \alpha \nabla \log p(x_t) + \sqrt{2\eta}z_{t+1}\]

where \(z_{t+1} \sim \mathcal{N}(0,I)\) is a standard Gaussian noise vector.

Parameters:

  • X (Dict) – Dictionary containing the current state \(x_t\).

  • y (torch.Tensor) – Observed measurements/data tensor

  • physics (Physics) – Forward operator \(A\) that models the measurement process

  • cur_data_fidelity (DataFidelity) – Negative log-likelihood function

  • cur_prior (ScorePrior) – Score-based prior model for \(\nabla \log p(x)\)

  • iteration (int) – Current iteration number in the sampling process (zero-indexed)

Returns:

Dictionary {"x": x} containing the next state \(x_{t+1}\) in the Markov chain.

Return type:

Dict

Examples using ULAIterator:#

Building your custom MCMC sampling algorithm.

Building your custom MCMC sampling algorithm.

Uncertainty quantification with PnP-ULA.

Uncertainty quantification with PnP-ULA.