class sklearn.covariance.GraphicalLasso(alpha=0.01, *, mode='cd', covariance=None, tol=0.0001, enet_tol=0.0001, max_iter=100, verbose=False, eps=np.float64(2.220446049250313e-16), assume_centered=False)[source]#

Sparse inverse covariance estimation with an l1-penalized estimator.

Read more in the User Guide.

Changed in version v0.20: GraphLasso has been renamed to GraphicalLasso

alphafloat, default=0.01

The regularization parameter: the higher alpha, the more regularization, the sparser the inverse covariance. Range is (0, inf].

mode{‘cd’, ‘lars’}, default=’cd’

The Lasso solver to use: coordinate descent or LARS. Use LARS for very sparse underlying graphs, where p > n. Elsewhere prefer cd which is more numerically stable.

covariance“precomputed”, default=None

If covariance is “precomputed”, the input data in fit is assumed to be the covariance matrix. If None, the empirical covariance is estimated from the data X.

Added in version 1.3.

tolfloat, default=1e-4

The tolerance to declare convergence: if the dual gap goes below this value, iterations are stopped. Range is (0, inf].

enet_tolfloat, default=1e-4

The tolerance for the elastic net solver used to calculate the descent direction. This parameter controls the accuracy of the search direction for a given column update, not of the overall parameter estimate. Only used for mode=’cd’. Range is (0, inf].

max_iterint, default=100

The maximum number of iterations.

verbosebool, default=False

If verbose is True, the objective function and dual gap are plotted at each iteration.

epsfloat, default=eps

The machine-precision regularization in the computation of the Cholesky diagonal factors. Increase this for very ill-conditioned systems. Default is np.finfo(np.float64).eps.

Added in version 1.3.

assume_centeredbool, default=False

If True, data are not centered before computation. Useful when working with data whose mean is almost, but not exactly zero. If False, data are centered before computation.

location_ndarray of shape (n_features,)

Estimated location, i.e. the estimated mean.

covariance_ndarray of shape (n_features, n_features)

Estimated covariance matrix

precision_ndarray of shape (n_features, n_features)

Estimated pseudo inverse matrix.


Number of iterations run.

costs_list of (objective, dual_gap) pairs

The list of values of the objective function and the dual gap at each iteration. Returned only if return_costs is True.

Added in version 1.3.


Number of features seen during fit.

Added in version 0.24.

feature_names_in_ndarray of shape (n_features_in_,)

Names of features seen during fit. Defined only when X has feature names that are all strings.

Added in version 1.0.

See also


L1-penalized covariance estimator.


Sparse inverse covariance with cross-validated choice of the l1 penalty.


>>> import numpy as np
>>> from sklearn.covariance import GraphicalLasso
>>> true_cov = np.array([[0.8, 0.0, 0.2, 0.0],
...                      [0.0, 0.4, 0.0, 0.0],
...                      [0.2, 0.0, 0.3, 0.1],
...                      [0.0, 0.0, 0.1, 0.7]])
>>> np.random.seed(0)
>>> X = np.random.multivariate_normal(mean=[0, 0, 0, 0],
...                                   cov=true_cov,
...                                   size=200)
>>> cov = GraphicalLasso().fit(X)
>>> np.around(cov.covariance_, decimals=3)
array([[0.816, 0.049, 0.218, 0.019],
       [0.049, 0.364, 0.017, 0.034],
       [0.218, 0.017, 0.322, 0.093],
       [0.019, 0.034, 0.093, 0.69 ]])
>>> np.around(cov.location_, decimals=3)
array([0.073, 0.04 , 0.038, 0.143])
error_norm(comp_cov, norm='frobenius', scaling=True, squared=True)[source]#

Compute the Mean Squared Error between two covariance estimators.

comp_covarray-like of shape (n_features, n_features)

The covariance to compare with.

norm{“frobenius”, “spectral”}, default=”frobenius”

The type of norm used to compute the error. Available error types: - ‘frobenius’ (default): sqrt(tr(A^t.A)) - ‘spectral’: sqrt(max(eigenvalues(A^t.A)) where A is the error (comp_cov - self.covariance_).

scalingbool, default=True

If True (default), the squared error norm is divided by n_features. If False, the squared error norm is not rescaled.

squaredbool, default=True

Whether to compute the squared error norm or the error norm. If True (default), the squared error norm is returned. If False, the error norm is returned.


The Mean Squared Error (in the sense of the Frobenius norm) between self and comp_cov covariance estimators.

fit(X, y=None)[source]#

Fit the GraphicalLasso model to X.

Xarray-like of shape (n_samples, n_features)

Data from which to compute the covariance estimate.


Not used, present for API consistency by convention.


Returns the instance itself.


Get metadata routing of this object.

Please check User Guide on how the routing mechanism works.


A MetadataRequest encapsulating routing information.


Get parameters for this estimator.

deepbool, default=True

If True, will return the parameters for this estimator and contained subobjects that are estimators.


Parameter names mapped to their values.


Getter for the precision matrix.

precision_array-like of shape (n_features, n_features)

The precision matrix associated to the current covariance object.


Compute the squared Mahalanobis distances of given observations.

Xarray-like of shape (n_samples, n_features)

The observations, the Mahalanobis distances of the which we compute. Observations are assumed to be drawn from the same distribution than the data used in fit.

distndarray of shape (n_samples,)

Squared Mahalanobis distances of the observations.

score(X_test, y=None)[source]#

Compute the log-likelihood of X_test under the estimated Gaussian model.

The Gaussian model is defined by its mean and covariance matrix which are represented respectively by self.location_ and self.covariance_.

X_testarray-like of shape (n_samples, n_features)

Test data of which we compute the likelihood, where n_samples is the number of samples and n_features is the number of features. X_test is assumed to be drawn from the same distribution than the data used in fit (including centering).


Not used, present for API consistency by convention.


The log-likelihood of X_test with self.location_ and self.covariance_ as estimators of the Gaussian model mean and covariance matrix respectively.


Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as Pipeline). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.


Estimator parameters.

selfestimator instance

Estimator instance.

set_score_request(*, X_test: bool | None | str = '$UNCHANGED$') GraphicalLasso[source]#

Request metadata passed to the score method.

Note that this method is only relevant if enable_metadata_routing=True (see sklearn.set_config). Please see User Guide on how the routing mechanism works.

The options for each parameter are:

  • True: metadata is requested, and passed to score if provided. The request is ignored if metadata is not provided.

  • False: metadata is not requested and the meta-estimator will not pass it to score.

  • None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.

  • str: metadata should be passed to the meta-estimator with this given alias instead of the original name.

The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.

Added in version 1.3.


This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a Pipeline. Otherwise it has no effect.

X_teststr, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED

Metadata routing for X_test parameter in score.


The updated object.