pylops.optimization.sparsity.IRLS

pylops.optimization.sparsity.IRLS(Op, data, nouter, threshR=False, epsR=1e-10, epsI=1e-10, x0=None, tolIRLS=1e-10, returnhistory=False, kind='data', **kwargs_solver)[source]

Iteratively reweighted least squares.

Solve an optimization problem with \(L^1\) cost function (data IRLS) or \(L^1\) regularization term (model IRLS) given the operator Op and data y.

In the data IRLS, the cost function is minimized by iteratively solving a weighted least squares problem with the weight at iteration \(i\) being based on the data residual at iteration \(i-1\). This IRLS solver is robust to outliers since the L1 norm given less weight to large residuals than L2 norm does.

Similarly in the model IRLS, the weight at at iteration \(i\) is based on the model at iteration \(i-1\). This IRLS solver inverts for a sparse model vector.

Parameters:
Op : pylops.LinearOperator

Operator to invert

data : numpy.ndarray

Data

nouter : int

Number of outer iterations

threshR : bool, optional

Apply thresholding in creation of weight (True) or damping (False)

epsR : float, optional

Damping to be applied to residuals for weighting term

espI : float, optional

Tikhonov damping

x0 : numpy.ndarray, optional

Initial guess

tolIRLS : float, optional

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

returnhistory : bool, optional

Return history of inverted model for each outer iteration of IRLS

kind : str, optional

Kind of solver (data or model)

**kwargs_solver

Arbitrary keyword arguments for scipy.sparse.linalg.cg solver for data IRLS and scipy.sparse.linalg.lsqr solver for model IRLS when using numpy data(or pylops.optimization.solver.cg and pylops.optimization.solver.cgls when using cupy data)

Returns:
xinv : numpy.ndarray

Inverted model

nouter : int

Number of effective outer iterations

xinv_hist : numpy.ndarray, optional

History of inverted model

rw_hist : numpy.ndarray, optional

History of weights

Notes

Data IRLS solves the following optimization problem for the operator \(\mathbf{Op}\) and the data \(\mathbf{d}\):

\[J = \|\mathbf{d} - \mathbf{Op}\,\mathbf{x}\|_1\]

by a set of outer iterations which require to repeatedly solve a weighted least squares problem of the form:

\[\DeclareMathOperator*{\argmin}{arg\,min} \mathbf{x}^{(i+1)} = \argmin_\mathbf{x} \|\mathbf{d} - \mathbf{Op}\,\mathbf{x}\|_{2, \mathbf{R}^{(i)}}^2 + \epsilon_\mathbf{I}^2 \|\mathbf{x}\|_2^2\]

where \(\mathbf{R}^{(i)}\) is a diagonal weight matrix whose diagonal elements at iteration \(i\) are equal to the absolute inverses of the residual vector \(\mathbf{r}^{(i)} = \mathbf{y} - \mathbf{Op}\,\mathbf{x}^{(i)}\) at iteration \(i\). More specifically the \(j\)-th element of the diagonal of \(\mathbf{R}^{(i)}\) is

\[R^{(i)}_{j,j} = \frac{1}{\left| r^{(i)}_j \right| + \epsilon_\mathbf{R}}\]

or

\[R^{(i)}_{j,j} = \frac{1}{\max\{\left|r^{(i)}_j\right|, \epsilon_\mathbf{R}\}}\]

depending on the choice threshR. In either case, \(\epsilon_\mathbf{R}\) is the user-defined stabilization/thresholding factor [1].

Similarly model IRLS solves the following optimization problem for the operator \(\mathbf{Op}\) and the data \(\mathbf{d}\):

\[J = \|\mathbf{x}\|_1 \quad \text{subject to} \quad \mathbf{d} = \mathbf{Op}\,\mathbf{x}\]

by a set of outer iterations which require to repeatedly solve a weighted least squares problem of the form [2]:

\[\mathbf{x}^{(i+1)} = \operatorname*{arg\,min}_\mathbf{x} \|\mathbf{x}\|_{2, \mathbf{R}^{(i)}}^2 \quad \text{subject to} \quad \mathbf{d} = \mathbf{Op}\,\mathbf{x}\]

where \(\mathbf{R}^{(i)}\) is a diagonal weight matrix whose diagonal elements at iteration \(i\) are equal to the absolutes of the model vector \(\mathbf{x}^{(i)}\) at iteration \(i\). More specifically the \(j\)-th element of the diagonal of \(\mathbf{R}^{(i)}\) is

\[R^{(i)}_{j,j} = \left|x^{(i)}_j\right|.\]
[1]https://en.wikipedia.org/wiki/Iteratively_reweighted_least_squares
[2]Chartrand, R., and Yin, W. “Iteratively reweighted algorithms for compressive sensing”, IEEE. 2008.

Examples using pylops.optimization.sparsity.IRLS