# Optimization¶

The pygsp.optimization module provides tools for convex optimization on graphs.

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

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) sol: solution