# pylops.optimization.cls_sparsity.SPGL1¶

class pylops.optimization.cls_sparsity.SPGL1(Op, callbacks=None)[source]

Solve a constrained system of equations given the operator Op and a sparsyfing transform SOp 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 the spgl1 library.

Parameters
Oppylops.LinearOperator

Operator to invert of size $$[N \times M]$$.

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