pylops.optimization.cls_sparsity.SPGL1#
- class pylops.optimization.cls_sparsity.SPGL1(Op, callbacks=None)[source]#
Spectral Projected-Gradient for L1 norm.
Solve a constrained system of equations given the operator
Op
and a sparsyfing transformSOp
aiming to retrive a model that is sparse in the sparsifying domain.This is a simple wrapper to
spgl1.spgl1
which is a porting of the well-known SPGL1 MATLAB solver into Python. In order to be able to use this solver you need to have installed thespgl1
library.- Parameters
- Op
pylops.LinearOperator
Operator to invert of size \([N \times M]\).
- Op
- Raises
- ModuleNotFoundError
If the
spgl1
library is not installed
Notes
Solve different variations of sparsity-promoting inverse problem by imposing sparsity in the retrieved model [1].
The first problem is called basis pursuit denoise (BPDN) and its cost function is
\[\|\mathbf{x}\|_1 \quad \text{subject to} \quad \left\|\mathbf{Op}\,\mathbf{S}^H\mathbf{x}-\mathbf{y}\right\|_2^2 \leq \sigma,\]while the second problem is the ℓ₁-regularized least-squares or LASSO problem and its cost function is
\[\left\|\mathbf{Op}\,\mathbf{S}^H\mathbf{x}-\mathbf{y}\right\|_2^2 \quad \text{subject to} \quad \|\mathbf{x}\|_1 \leq \tau\]- 1
van den Berg E., Friedlander M.P., “Probing the Pareto frontier for basis pursuit solutions”, SIAM J. on Scientific Computing, vol. 31(2), pp. 890-912. 2008.
Methods
__init__
(Op[, callbacks])callback
(x, *args, **kwargs)Callback routine
finalize
(*args[, show])Finalize solver
run
(x[, show])Run solver
setup
(y[, SOp, tau, sigma, show])Setup solver
solve
(y[, x0, SOp, tau, sigma, show])Run entire solver
step
()Run one step of solver