pyproximal.optimization.pnp.PlugAndPlay#
- pyproximal.optimization.pnp.PlugAndPlay(proxf, denoiser, dims, x0, tau, niter=10, callback=None, show=False)[source]#
Plug-and-Play Priors with ADMM optimization
Solves the following minimization problem using the ADMM algorithm:
\[\mathbf{x},\mathbf{z} = \argmin_{\mathbf{x}} f(\mathbf{x}) + \lambda g(\mathbf{x})\]where \(f(\mathbf{x})\) is a function that has a known proximal operator where \(g(\mathbf{x})\) is a function acting as implicit prior. Implicit means that no explicit function should be defined: instead, a denoising algorithm of choice is used. See Notes for details.
- Parameters
- proxf
pyproximal.ProxOperator Proximal operator of f function
- denoiser
func Denoiser (must be a function with two inputs, the first is the signal to be denoised, the second is the tau constant of the y-update in PlugAndPlay)
- dims
tuple Dimensions used to reshape the vector
xin theproxmethod prior to calling thedenoiser- x0
numpy.ndarray Initial vector
- tau
float, optional Positive scalar weight, which should satisfy the following condition to guarantees convergence: \(\tau \in (0, 1/L]\) where
Lis the Lipschitz constant of \(\nabla f\).- niter
int, optional Number of iterations of iterative scheme
- callback
callable, optional Function with signature (
callback(x)) to call after each iteration wherexis the current model vector- show
bool, optional Display iterations log
- proxf
- Returns
- x
numpy.ndarray Inverted model
- z
numpy.ndarray Inverted second model
- x
Notes
Plug-and-Play Priors [1] can be expressed by the following recursion:
\[\begin{split}\mathbf{x}^{k+1} = \prox_{\tau f}(\mathbf{z}^{k} - \mathbf{u}^{k})\\ \mathbf{z}^{k+1} = \operatorname{Denoise}(\mathbf{x}^{k+1} + \mathbf{u}^{k}, \tau \lambda)\\ \mathbf{u}^{k+1} = \mathbf{u}^{k} + \mathbf{x}^{k+1} - \mathbf{z}^{k+1}\end{split}\]where \(\operatorname{Denoise}\) is a denoising algorithm of choice. This rather peculiar step originates from the intuition that the optimization process associated with the z-update can be interpreted as a denoising inverse problem, or more specifically a MAP denoiser where the noise is gaussian with zero mean and variance equal to \(\tau \lambda\). For this reason any denoising of choice can be used instead of a function with known proximal operator.
Finally, whilst the \(\tau \lambda\) denoising parameter should be chosen to represent an estimate of the noise variance (of the denoiser, not the data of the problem we wish to solve!), special care must be taken when setting up the denoiser and calling this optimizer. More specifically, \(\lambda\) should not be passed to the optimizer, rather set directly in the denoiser. On the other hand \(\tau\) must be passed to the optimizer as it is also affecting the x-update; when defining the denoiser, ensure that \(\tau\) is multiplied to \(\lambda\) as shown in the tutorial.
Alternative, as suggested in [2], the \(\tau\) could be set to 1. The parameter \(\lambda\) can then be set to maximize the value of the denoiser and a second tuning parameter can be added directly to \(f\).
