MoreauYoshidaPrior

class cuqi.implicitprior.MoreauYoshidaPrior(prior, smoothing_strength=0.1, **kwargs)

This class defines (implicit) smoothed priors for which we can apply gradient-based algorithms. The smoothing is performed using the Moreau-Yoshida envelope of the target prior potential.

In the following we give a detailed explanation of the Moreau-Yoshida smoothing.

We consider a density such that - logpi(x) = -g(x) with g convex, lsc, proper but not differentiable. Consequently, we cannot apply any algorithm requiring the gradient of g. Idea: We consider the Moreau envelope of g defined as

g_{smoothing_strength} (x) = inf_z 0.5*| x-z |_2^2/smoothing_strength + g(z).

g_{smoothing_strength} has some nice properties

• g_{smoothing_strength}(x)–>g(x) as smoothing_strength–>0 for all x

abla g_{smoothing_strength} is 1/smoothing_strength-Lipschitz

abla g_{smoothing_strength}(x) = (x - prox_g^{smoothing_strength}(x))/smoothing_strength for all x with

prox_g^{smoothing_strength}(x) = argmin_z 0.5*| x-z |_2^2/smoothing_strength + g(z) .

Consequently, we can apply any gradient-based algorithm with g_{smoothing_strength} in lieu of g. These algorithms do not require the full knowledge of g_{smoothing_strength} but only its gradient. The gradient of g_{smoothing_strength} is fully determined by prox_g^{smoothing_strength} and smoothing_strength. It is important as, although there exists an explicit formula for g_{smoothing_strength}, it is rarely used in practice, as it would require us to solve an optimization problem each time we want to estimate g_{smoothing_strength}. Furthermore, there exist cases where we dont’t the regularization g with which the mapping prox_g^{smoothing_strength} is associated.

Remark (Proximal operators are denoisers):

We consider the denoising inverse problem x = u + n, with n sim mathcal{N}(0, smoothing_strength I). A mapping solving a denoising inverse problem is called denoiser. It takes the noisy observation x as an input and returns a less noisy version of x which is an estimate of u. We assume a prior density pi(u) propto exp(- g(u)). Then the MAP estimate is given by

x_MAP = rgmin_z 0.5 | x - z |_2^2/smoothing_strength + g(z) = prox_g^smoothing_strength(x)

Then proximal operators are denoisers.

Remark (Denoisers are not necessarily proximal operators): Data-driven denoisers are not necessarily proximal operators (see https://arxiv.org/pdf/2201.13256)

priorRestorationPrior

Prior of the RestorationPrior type. In order to stay within the MYULA framework the restorator of RestorationPrior must be a proximal operator.

smoothing_strengthfloat

Smoothing strength of the Moreau-Yoshida envelope of the prior potential.

__init__(prior, smoothing_strength=0.1, **kwargs)

Initialize the core properties of the distribution.

Parameters:

• name (str, default None) – Name of distribution.

• geometry (Geometry, default _DefaultGeometry (or None)) – Geometry of distribution.

• is_symmetric (bool, default None) – Indicator if distribution is symmetric.

Methods

__init__(prior[, smoothing_strength])	Initialize the core properties of the distribution.
disable_FD()	Disable finite difference approximation for logd gradient.
enable_FD([epsilon])	Enable finite difference approximation for logd gradient.
get_conditioning_variables()	Returns the conditioning variables of the distribution.
get_mutable_variables()	Return any public variable that is mutable (attribute or property) except those in the ignore_vars list
get_parameter_names()	Returns the names of the parameters that the density can be evaluated at or conditioned on.
gradient(x)	This is the gradient of the regularizer ie gradient of the negative logpdf of the implicit prior.
logd(*args, **kwargs)	Evaluate the un-normalized log density function of the distribution.
logpdf(x)	The logpdf function.
pdf(x)	Evaluate the log probability density function of the distribution.
sample([N])	Sample from the distribution.
to_likelihood(data)	Convert conditional distribution to a likelihood function given observed data

Attributes

FD_enabled	Returns True if finite difference approximation of the logd gradient is enabled.
FD_epsilon	Spacing for the finite difference approximation of the logd gradient.
dim	Return the dimension of the distribution based on the geometry.
geometry	Return the geometry of the distribution.
is_cond	Returns True if instance (self) is a conditional distribution.
name	Name of the random variable associated with the density.
prior	Getter for the MoreauYoshida prior.
smoothing_strength	smoothing_strength of the distribution