ULA#

class deepinv.sampling.ULA(prior, data_fidelity, step_size=1.0, sigma=0.05, alpha=1.0, max_iter=1e3, thinning=5, burnin_ratio=0.2, clip=(-1.0, 2.0), thresh_conv=1e-3, save_chain=False, verbose=False)[source]#

Bases: BaseSampling

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 required step_size smaller than \(\frac{1}{L+\|A\|_2^2}\)

Warning

This a legacy class provided for convenience. See the example in Markov Chain Monte Carlo for details on how to build a ULA sampler.

Parameters:

  • prior (deepinv.optim.ScorePrior, torch.nn.Module) – negative log-prior based on a trained or model-based denoiser.

  • data_fidelity (deepinv.optim.DataFidelity, torch.nn.Module) – negative log-likelihood function linked with the noise distribution in the acquisition physics.

  • step_size (float) – step size \(\eta>0\) of the algorithm. Tip: use deepinv.physics.LinearPhysics.compute_norm() to compute the Lipschitz constant of a linear forward operator.

  • sigma (float) – noise level used in the plug-and-play prior denoiser. A larger value of sigma will result in a more regularized reconstruction.

  • alpha (float) – regularization parameter \(\alpha\)

  • max_iter (int) – number of Monte Carlo iterations.

  • thinning (int) – Thins the Markov Chain by an integer \(\geq 1\) (i.e., keeping one out of thinning samples to compute posterior statistics).

  • burnin_ratio (float) – percentage of iterations used for burn-in period, should be set between 0 and 1. The burn-in samples are discarded constant with a numerical algorithm.

  • clip (tuple) – Tuple containing the box-constraints \([a,b]\). If None, the algorithm will not project the samples.

  • crit_conv (float) – Threshold for verifying the convergence of the mean and variance estimates.

  • verbose (bool) – prints progress of the algorithm.

forward(y, physics, seed=None, x_init=None, g_statistics=lambda d: ...)[source]#

Runs the chain to obtain the posterior mean and variance of the reconstruction of the measurements y.

Parameters:

  • y (torch.Tensor) – Measurements

  • physics (deepinv.physics.Physics) – Forward operator associated with the measurements

  • seed (float) – Random seed for generating the Monte Carlo samples

  • g_statistics (list[Callable] | Callable) – List of functions for which to compute posterior statistics, or a single function. The sampler will compute the posterior mean and variance of each function in the list. Note the sampler outputs a dictionary so they must act on d["x"]. Default: lambda d: d["x"] (identity function)

Returns:

(tuple of torch.tensor) containing the posterior mean and variance.

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