Class AugmentedLinearRegressionL1L2TV

source code

                      object --+        
                               |        
             properties.Function --+    
                                   |    
       properties.SplittableFunction --+
                                       |
                      object --+       |
                               |       |
     properties.ProximalOperator --+   |
                                   |   |
properties.AugmentedProximalOperator --+
                                       |
                                      AugmentedLinearRegressionL1L2TV

Combination (sum) of LinearRegression, L1, L2 and 1D TotalVariation
with a linear constraint. Represents the problem

    min. f(b) = g(x)
                + h(r)
              = (1 / 2) ||Xb - y||² + (k / 2) ||b||²
                + l ||r_1||_1 + g ||r_tv||_1,
    s.t. r = Db.

The proximal operators of the splittable functions are assumed to be from
augmented Lagrangians.

Note: This function only works for 1-dimensional total variation.

Parameters
----------
X : Numpy array. The X matrix for the linear regression.

y : Numpy array. The y vector for the linear regression.

l1 : Non-negative float. The Lagrange multiplier, or regularisation
        constant, for the L1 penalty.

l2 : Non-negative float. The Lagrange multiplier, or regularisation
        constant, for the ridge penalty.

tv : Non-negative float. The Lagrange multiplier, or regularisation
        constant, of the total variation function.

A : List or tuple of numpy (or usually sparse scipy) arrays. The linear
        operator for the constraints.

rho : Positive float. The penalty parameter for the augmented Lagrangian.

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

mean : Boolean. Whether to compute the squared loss or the mean
        squared loss. Default is True, the mean squared loss.

The proximal operator corresponding to the function.

Parameters
----------
x : Numpy array (p-by-1). The point at which to apply the proximal
        operator.

factor : Positive float. A factor by which the Lagrange multiplier is
        scaled. This is usually the step size.

eps : Positive float. This is the stopping criterion for inexact
        proximal methods, where the proximal operator is approximated
        numerically.

max_iter : Positive integer. This is the maximum number of iterations
        for inexact proximal methods, where the proximal operator is
        approximated numerically.

index : Non-negative integer. For multivariate functions, this
        identifies the variable for which the proximal operator is
        associated.