Optimization

The pygsp.optimization module provides tools to solve convex optimization problems on graphs.

pygsp.optimization.prox_tv(x, gamma, G, A=None, At=None, nu=1, tol=0.001, maxit=200, use_matrix=True)

Total Variation proximal operator for graphs.

This function computes the TV proximal operator for graphs. The TV norm is the one norm of the gradient. The gradient is defined in the function pygsp.graphs.Graph.grad(). This function requires the PyUNLocBoX to be executed.

This function solves:

\(sol = \min_{z} \frac{1}{2} \|x - z\|_2^2 + \gamma \|x\|_{TV}\)

Parameters

x: int

Input signal

gamma: ndarray

Regularization parameter

G: graph object

Graphs structure

A: lambda function

Forward operator, this parameter allows to solve the following problem: \(sol = \min_{z} \frac{1}{2} \|x - z\|_2^2 + \gamma \| A x\|_{TV}\) (default = Id)

At: lambda function

Adjoint operator. (default = Id)

nu: float

Bound on the norm of the operator (default = 1)

tol: float

Stops criterion for the loop. The algorithm will stop if : \(\frac{n(t) - n(t - 1)} {n(t)} < tol\) where \(n(t) = f(x) + 0.5 \|x-y\|_2^2\) is the objective function at iteration \(t\) (default = \(10e-4\))

maxit: int

Maximum iteration. (default = 200)

use_matrix: bool

If a matrix should be used. (default = True)

Returns

sol: solution