pylops.optimization.sparsity.FISTA(Op, data, niter, eps=0.1, alpha=None, eigsiter=None, eigstol=0, tol=1e-10, returninfo=False, show=False, threshkind='soft', perc=None, callback=None, decay=None, SOp=None, x0=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 \(N<M\).

Op : pylops.LinearOperator

Operator to invert

data : numpy.ndarray


niter : int

Number of iterations

eps : float, optional

Sparsity damping Step size. To guarantee convergence, ensure \(\alpha \le 1/\lambda_\text{max}\), where \(\lambda_\text{max}\) is the largest eigenvalue of \(\mathbf{Op}^H\mathbf{Op}\). If None, the maximum eigenvalue is estimated and the optimal step size is chosen as \(1/\lambda_\text{max}\). If provided, the convergence criterion will not be checked internally.

eigsiter : int, optional

Number of iterations for eigenvalue estimation if alpha=None

eigstol : float, optional

Tolerance for eigenvalue estimation if alpha=None

tol : float, optional

Tolerance. Stop iterations if difference between inverted model at subsequent iterations is smaller than tol

returninfo : bool, optional

Return info of FISTA solver

show : bool, optional

Display iterations log

threshkind : str, optional

Kind of thresholding (‘hard’, ‘soft’, ‘half’, ‘soft-percentile’, or ‘half-percentile’ - ‘soft’ used as default)

perc : float, optional

Percentile, as percentage of values to be kept by thresholding (to be provided when thresholding is soft-percentile or half-percentile)

callback : callable, optional

Function with signature (callback(x)) to call after each iteration where x is the current model vector

decay : numpy.ndarray, optional

Decay factor to be applied to thresholding during iterations

SOp : pylops.LinearOperator, optional

Regularization operator (use when solving the analysis problem)

x0: :obj:`numpy.ndarray`, optional

Initial guess

xinv : numpy.ndarray

Inverted model

niter : int

Number of effective iterations

cost : numpy.ndarray, optional

History of cost function


If threshkind is different from hard, soft, half, soft-percentile, or half-percentile


If perc=None when threshkind is soft-percentile or half-percentile

See also

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


Solves the following synthesis problem for the operator \(\mathbf{Op}\) and the data \(\mathbf{d}\):

\[J = \|\mathbf{d} - \mathbf{Op}\,\mathbf{x}\|_2^2 + \epsilon \|\mathbf{x}\|_p\]

or the analysis problem:

\[J = \|\mathbf{d} - \mathbf{Op}\,\mathbf{x}\|_2^2 + \epsilon \|\mathbf{SOp}^H\,\mathbf{x}\|_p\]

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.