Package parsimony :: Package functions :: Package nesterov :: Module l1 :: Class L1

Class L1

                            object --+        
                                     |        
                   properties.Gradient --+    
                                         |    
                            object --+   |    
                                     |   |    
properties.LipschitzContinuousGradient --+    
                                         |    
                            object --+   |    
                                     |   |    
                properties.Eigenvalues --+    
                                         |    
                            object --+   |    
                                     |   |    
           properties.ProximalOperator --+    
                                         |    
               properties.NesterovFunction --+
                                             |
                                object --+   |
                                         |   |
                        properties.Penalty --+
                                             |
                                object --+   |
                                         |   |
                     properties.Constraint --+
                                             |
                                            L1

The proximal operator of the smoothed L1 function

    f(beta) = l * (L1mu(beta) - c),

where L1mu(eta) is the smoothed L1 function. The constrained version has
the form

    L1mu(beta) <= c.

Instance Methods

[hide private]

__init__(self, l, c=0.0, A=None, mu=0.0, penalty_start=0)
Parameters ---------- l : Non-negative float.

source code

f(self, beta)
Function value.

source code

fmu(self, beta, mu=None)
Returns the smoothed function value.

source code

phi(self, alpha, beta)
Function value with known alpha.

source code

grad(self, beta)
Gradient of the function at beta.

source code

L(self)
Lipschitz constant of the gradient.

source code

alpha(self, beta)
Dual variable of the Nesterov function.

source code

M(self)
The maximum value of the regularisation of the dual variable.

source code

estimate_mu(self, beta)
Computes a "good" value of mu with respect to the given eta.

source code

feasible(self, beta)
Feasibility of the constraint.

source code

Inherited from properties.NesterovFunction: A, Aa, get_mu, lA, lambda_max, prox, set_mu

Inherited from properties.Gradient: approx_grad

Inherited from properties.LipschitzContinuousGradient: approx_L

Inherited from properties.Eigenvalues: lambda_min

Inherited from object: __delattr__, __format__, __getattribute__, __hash__, __new__, __reduce__, __reduce_ex__, __repr__, __setattr__, __sizeof__, __str__, __subclasshook__

Static Methods

[hide private]

project(a)
Projection onto the compact space of the Nesterov function.

source code

Class Variables

[hide private]

__abstractmethods__ = frozenset([])

Inherited from properties.NesterovFunction: __metaclass__

Inherited from properties.Gradient (private): _abc_cache, _abc_negative_cache, _abc_negative_cache_version, _abc_registry

Properties

[hide private]

Inherited from object: __class__

Method Details

[hide private]

init(self, l, c=0.0, A=None, mu=0.0, penalty_start=0)
(Constructor)

source code


Parameters
----------
l : Non-negative float. The Lagrange multiplier, or regularisation
        constant, of the function.

c : Float. The limit of the constraint. The function is feasible if
        ||beta||_1 <= c. The default value is c=0, i.e. the default is
        a regularisation formulation.

A : A (usually sparse) matrix. The linear operator for the Nesterov
        formulation. May not be None.

mu : Non-negative float. The regularisation constant for the smoothing.

penalty_start : Non-negative integer. The number of columns, variables
        etc., to exempt from penalisation. Equivalently, the first
        index to be penalised. Default is 0, all columns are included.

Overrides: object.__init__

fmu(self, beta, mu=None)

source code

Returns the smoothed function value.

Parameters: ---------- beta : A weight vector.

mu : The regularisation constant for the smoothing.

Overrides: properties.NesterovFunction.fmu

phi(self, alpha, beta)

source code

Function value with known alpha.

From the interface "NesterovFunction".

Overrides: properties.NesterovFunction.phi

grad(self, beta)

source code

Gradient of the function at beta.

From the interface "Gradient". Overloaded since we can do it faster than the default.

Overrides: properties.Gradient.grad

L(self)

source code

Lipschitz constant of the gradient.

From the interface "LipschitzContinuousGradient".

Overrides: properties.LipschitzContinuousGradient.L

alpha(self, beta)

source code

Dual variable of the Nesterov function.

From the interface "NesterovFunction". Overloaded since we can do it faster than the default.

Overrides: properties.NesterovFunction.alpha

project(a)
Static Method

source code

Projection onto the compact space of the Nesterov function.

From the interface "NesterovFunction".

Overrides: properties.NesterovFunction.project

M(self)

source code

The maximum value of the regularisation of the dual variable. We
have

    M = max_{alpha in K} 0.5*|alpha|²_2.

From the interface "NesterovFunction".