  We develop a method to evaluate the generalized degrees of freedom (GDF),
which is a key quantity of a model selection criterion, for linear regression
with sparse regularization. Using the replica method, GDF is expressed by the
variables that characterize the saddle point of the free energy without
depending on the form of the regularization. Within the framework of replica
symmetric (RS) analysis, GDF is provided with a physical meaning as the
effective density of non-zero components. The validity of our method in the RS
phase is supported by the consistency of our results with previous mathematical
results. The analytical results in the RS phase are numerically achieved by the
belief propagation algorithm.
