ProxNest.optimisations.l1_norm_prox.l1_norm_prox(x, lamb, params)

Proximal operator associated with L1 norm.

Compute the L1 proximal operator, i.e. solve

\[z^* = \min_{z} \frac{1}{2}||x - z||_2^2 + \lambda * ||\Psi^{\dagger} z||_1,\]

where \(x\) is the input vector and the solution \(z^*\) is returned as sol.

Parameters

• x (np.ndarray) – A sample position \(x\) in the posterior space.

• lamb (float) – Regularisation parameter.

• params (dict) – Dictionary of parameters defining the optimisation.

Returns

Optimal solution \(z^*\) of proximal operator.

Return type

np.ndarray

Notes

[1] M.J. Fadili and J-L. Starck, “Monotone operator splitting for optimization problems in sparse recovery” , IEEE ICIP, Cairo, Egypt, 2009.

[2] Amir Beck and Marc Teboulle, “A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems”, SIAM Journal on Imaging Sciences 2 (2009), no. 1, 183–202.