pylops.optimization.sparsity.ISTA¶
-
pylops.optimization.sparsity.ISTA(Op, data, niter, eps=0.1, alpha=None, eigsiter=None, eigstol=0, tol=1e-10, monitorres=False, returninfo=False, show=False, threshkind='soft', perc=None, callback=None, decay=None, SOp=None, x0=None)[source]¶ Iterative Shrinkage-Thresholding Algorithm (ISTA).
Solve an optimization problem with \(L^p, \; p=0, 0.5, 1\) regularization, given the operator
Opand datay. 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
- data :
numpy.ndarray Data
- niter :
int Number of iterations
- eps :
float, optional Sparsity damping
- alpha :
float, optional 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 :
float, 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- monitorres :
bool, optional Monitor that residual is decreasing
- returninfo :
bool, optional Return info of CG solver
- show :
bool, optional Display iterations log
- threshkind :
str, optional Kind of thresholding (‘hard’, ‘soft’, ‘half’, ‘hard-percentile’, ‘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 wherexis 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
Returns: - xinv :
numpy.ndarray Inverted model
- niter :
int Number of effective iterations
- cost :
numpy.ndarray, optional History of cost function
Raises: - NotImplementedError
If
threshkindis different from hard, soft, half, soft-percentile, or half-percentile- ValueError
If
perc=Nonewhenthreshkindis soft-percentile or half-percentile- ValueError
If
monitorres=Trueand residual increases
See also
OMP- Orthogonal Matching Pursuit (OMP).
FISTA- Fast Iterative Shrinkage-Thresholding Algorithm (FISTA).
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 \(\mathbf{d}\):
\[J = \|\mathbf{d} - \mathbf{Op}\,\mathbf{x}\|_2^2 + \epsilon \|\mathbf{x}\|_p\]or the analysis problem:
\[J = \|\mathbf{d} - \mathbf{Op}\,\mathbf{x}\|_2^2 + \epsilon \|\mathbf{SOp}^H\,\mathbf{x}\|_p\]if
SOpis provided. Note that in the first case,SOpshould be assimilated in the modelling operator (i.e.,Op=GOp * SOp).The Iterative Shrinkage-Thresholding Algorithms (ISTA) [1] is used, where \(p=0, 0.5, 1\). This is a very simple iterative algorithm which applies the following step:
\[\mathbf{x}^{(i+1)} = T_{(\epsilon \alpha /2, p)} \left(\mathbf{x}^{(i)} + \alpha\,\mathbf{Op}^H \left(\mathbf{d} - \mathbf{Op}\,\mathbf{x}^{(i)}\right)\right)\]or
\[\mathbf{x}^{(i+1)} = \mathbf{SOp}\,\left\{T_{(\epsilon \alpha /2, p)} \mathbf{SOp}^H\,\left(\mathbf{x}^{(i)} + \alpha\,\mathbf{Op}^H \left(\mathbf{d} - \mathbf{Op} \,\mathbf{x}^{(i)}\right)\right)\right\}\]where \(\epsilon \alpha /2\) is the threshold and \(T_{(\tau, p)}\) is the thresholding rule. The most common variant of ISTA uses the so-called soft-thresholding rule \(T(\tau, p=1)\). Alternatively an hard-thresholding rule is used in the case of \(p=0\) or a half-thresholding rule is used in the case of \(p=1/2\). Finally, percentile bases thresholds are also implemented: the damping factor is not used anymore an the threshold changes at every iteration based on the computed percentile.
[1] Daubechies, I., Defrise, M., and De Mol, C., “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint”, Communications on pure and applied mathematics, vol. 57, pp. 1413-1457. 2004. - Op :
