lars_path_gram#

sklearn.linear_model.lars_path_gram(Xy, Gram, *, n_samples, max_iter=500, alpha_min=0, method='lar', copy_X=True, eps=np.float64(2.220446049250313e-16), copy_Gram=True, verbose=0, return_path=True, return_n_iter=False, positive=False)[source]#

The lars_path in the sufficient stats mode.

The optimization objective for the case method=’lasso’ is:

(1 / (2 * n_samples)) * ||y - Xw||^2_2 + alpha * ||w||_1

in the case of method=’lar’, the objective function is only known in the form of an implicit equation (see discussion in [1]).

Read more in the User Guide.

Parameters:

Xyndarray of shape (n_features,): Xy = X.T @ y.
Gramndarray of shape (n_features, n_features): Gram = X.T @ X.
n_samplesint: Equivalent size of sample.
max_iterint, default=500: Maximum number of iterations to perform, set to infinity for no limit.
alpha_minfloat, default=0: Minimum correlation along the path. It corresponds to the regularization parameter alpha parameter in the Lasso.
method{‘lar’, ‘lasso’}, default=’lar’: Specifies the returned model. Select 'lar' for Least Angle Regression, 'lasso' for the Lasso.
copy_Xbool, default=True: If False, X is overwritten.
epsfloat, default=np.finfo(float).eps: The machine-precision regularization in the computation of the Cholesky diagonal factors. Increase this for very ill-conditioned systems. Unlike the tol parameter in some iterative optimization-based algorithms, this parameter does not control the tolerance of the optimization.
copy_Grambool, default=True: If False, Gram is overwritten.
verboseint, default=0: Controls output verbosity.
return_pathbool, default=True: If return_path==True returns the entire path, else returns only the last point of the path.
return_n_iterbool, default=False: Whether to return the number of iterations.
positivebool, default=False: Restrict coefficients to be >= 0. This option is only allowed with method ‘lasso’. Note that the model coefficients will not converge to the ordinary-least-squares solution for small values of alpha. Only coefficients up to the smallest alpha value (alphas_[alphas_ > 0.].min() when fit_path=True) reached by the stepwise Lars-Lasso algorithm are typically in congruence with the solution of the coordinate descent lasso_path function.

Returns:

alphasndarray of shape (n_alphas + 1,): Maximum of covariances (in absolute value) at each iteration. n_alphas is either max_iter, n_features or the number of nodes in the path with alpha >= alpha_min, whichever is smaller.
activendarray of shape (n_alphas,): Indices of active variables at the end of the path.
coefsndarray of shape (n_features, n_alphas + 1): Coefficients along the path.
n_iterint: Number of iterations run. Returned only if return_n_iter is set to True.

See also

lars_path_gram: Compute LARS path.
lasso_path: Compute Lasso path with coordinate descent.
LassoLars: Lasso model fit with Least Angle Regression a.k.a. Lars.
Lars: Least Angle Regression model a.k.a. LAR.
LassoLarsCV: Cross-validated Lasso, using the LARS algorithm.
LarsCV: Cross-validated Least Angle Regression model.
sklearn.decomposition.sparse_encode: Sparse coding.

References

[1]

“Least Angle Regression”, Efron et al. http://statweb.stanford.edu/~tibs/ftp/lars.pdf

[2]

Wikipedia entry on the Least-angle regression

[3]

Wikipedia entry on the Lasso

Examples

>>> from sklearn.linear_model import lars_path_gram
>>> from sklearn.datasets import make_regression
>>> X, y, true_coef = make_regression(
...    n_samples=100, n_features=5, n_informative=2, coef=True, random_state=0
... )
>>> true_coef
array([ 0.        ,  0.        ,  0.        , 97.9..., 45.7...])
>>> alphas, _, estimated_coef = lars_path_gram(X.T @ y, X.T @ X, n_samples=100)
>>> alphas.shape
(3,)
>>> estimated_coef
array([[ 0.     ,  0.     ,  0.     ],
       [ 0.     ,  0.     ,  0.     ],
       [ 0.     ,  0.     ,  0.     ],
       [ 0.     , 46.96..., 97.99...],
       [ 0.     ,  0.     , 45.70...]])