# pylops.optimization.cls_sparsity.FISTA¶

class pylops.optimization.cls_sparsity.FISTA(Op, callbacks=None)[source]

Fast Iterative Shrinkage-Thresholding Algorithm (FISTA).

Solve an optimization problem with $$L_p, \; p=0, 0.5, 1$$ regularization, given the operator Op and data y. The operator can be real or complex, and should ideally be either square $$N=M$$ or underdetermined $$N<M$$.

Parameters: Op : pylops.LinearOperator Operator to invert NotImplementedError If threshkind is different from hard, soft, half, soft-percentile, or half-percentile ValueError If perc=None when threshkind is soft-percentile or half-percentile

OMP
Orthogonal Matching Pursuit (OMP).
ISTA
Iterative Shrinkage-Thresholding Algorithm (ISTA).
SPGL1
Spectral Projected-Gradient for L1 norm (SPGL1).
SplitBregman
Split Bregman for mixed L2-L1 norms.

Notes

Solves the following synthesis problem for the operator $$\mathbf{Op}$$ and the data $$\mathbf{y}$$:

$J = \|\mathbf{y} - \mathbf{Op}\,\mathbf{x}\|_2^2 + \epsilon \|\mathbf{x}\|_p$

or the analysis problem:

$J = \|\mathbf{y} - \mathbf{Op}\,\mathbf{x}\|_2^2 + \epsilon \|\mathbf{SOp}^H\,\mathbf{x}\|_p$

if SOp is provided.

The Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) [1] is used, where $$p=0, 0.5, 1$$. This is a modified version of ISTA solver with improved convergence properties and limited additional computational cost. Similarly to the ISTA solver, the choice of the thresholding algorithm to apply at every iteration is based on the choice of $$p$$.

 [1] Beck, A., and Teboulle, M., “A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems”, SIAM Journal on Imaging Sciences, vol. 2, pp. 183-202. 2009.

Methods

 __init__(Op[, callbacks]) Initialize self. callback(x, *args, **kwargs) Callback routine finalize([show]) Finalize solver run(x[, niter, show, itershow]) Run solver setup(y[, x0, niter, SOp, eps, alpha, …]) Setup solver solve(y[, x0, niter, SOp, eps, alpha, …]) Run entire solver step(x, z[, show]) Run one step of solver
step(x, z, show=False)[source]

Run one step of solver

Parameters: x : np.ndarray Current model vector to be updated by a step of ISTA x : np.ndarray Current auxiliary model vector to be updated by a step of ISTA show : bool, optional Display iteration log x : np.ndarray Updated model vector z : np.ndarray Updated auxiliary model vector xupdate : float Norm of the update
run(x, niter=None, show=False, itershow=[10, 10, 10])[source]

Run solver

Parameters: x : np.ndarray Current model vector to be updated by multiple steps of CG niter : int, optional Number of iterations. Can be set to None if already provided in the setup call show : bool, optional Display logs itershow : list, optional Display set log for the first N1 steps, last N2 steps, and every N3 steps in between where N1, N2, N3 are the three element of the list. x : np.ndarray Estimated model of size $$[M \times 1]$$