pylops.optimization.sparsity.SPGL1¶
-
pylops.optimization.sparsity.
SPGL1
(Op, data, SOp=None, tau=0, sigma=0, x0=None, **kwargs_spgl1)[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 sparsyfing 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
- data :
numpy.ndarray
Data
- SOp :
pylops.LinearOperator
Sparsyfing transform
- tau :
float
Non-negative LASSO scalar. If different from
0
, SPGL1 will solve LASSO problem- sigma :
list
BPDN scalar. If different from
0
, SPGL1 will solve BPDN problem- x0 :
numpy.ndarray
Initial guess
- **kwargs_spgl1
Arbitrary keyword arguments for
spgl1.spgl1
solver
Returns: - xinv :
numpy.ndarray
Inverted model in original domain.
- pinv :
numpy.ndarray
Inverted model in sparse domain.
- info :
dict
Dictionary with the following information:
tau
, final value of tau (see sigma above)rnorm
, two-norm of the optimal residualrgap
, relative duality gap (an optimality measure)gnorm
, Lagrange multiplier of (LASSO)stat
,1
: found a BPDN solution,2
: found a BP solution; exit based on small gradient,3
: found a BP solution; exit based on small residual,4
: found a LASSO solution,5
: error, too many iterations,6
: error, linesearch failed,7
: error, found suboptimal BP solution,8
: error, too many matrix-vector products.
niters
, number of iterationsnProdA
, number of multiplications with AnProdAt
, number of multiplications with A’n_newton
, number of Newton stepstime_project
, projection time (seconds)time_matprod
, matrix-vector multiplications time (seconds)time_total
, total solution time (seconds)niters_lsqr
, number of lsqr iterations (ifsubspace_min=True
)xnorm1
, L1-norm model solution history through iterationsrnorm2
, L2-norm residual history through iterationslambdaa
, Lagrange multiplier history through iterations
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
\[||\mathbf{x}||_1 \quad subj. to \quad ||\mathbf{Op}\mathbf{S}^H\mathbf{x}-\mathbf{b}||_2 <= \sigma,\]while the second problem is the l1-regularized least-squares or LASSO problem and its cost function is
\[||\mathbf{Op}\mathbf{S}^H\mathbf{x}-\mathbf{b}||_2 \quad subj. to \quad ||\mathbf{x}||_1 <= \tau\][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. - Op :