  This study proposes sparse estimation methods for the generalized linear
models, which run one of least angle regression (LARS) and least absolute
shrinkage and selection operator (LASSO) in the tangent space of the manifold
of the statistical model. This study approximates the statistical model and
subsequently uses exact calculations. LARS was proposed as an efficient
algorithm for parameter estimation and variable selection for the normal linear
model. The LARS algorithm is described in terms of Euclidean geometry regarding
the correlation as the metric of the parameter space. Since the LARS algorithm
only works in Euclidean space, we transform a manifold of the statistical model
into the tangent space at the origin. In the generalized linear regression,
this transformation allows us to run the original LARS algorithm for the
generalized linear models. The proposed methods are efficient and perform well.
Real-data analysis indicates that the proposed methods output similar results
to that of the $l_1$-regularized maximum likelihood estimation for the
aforementioned models. Numerical experiments reveal that our methods work well
and they may be better than the $l_1$-regularization in generalization,
parameter estimation, and model selection.
