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We develop a method to evaluate the generalized degrees of freedom (GDF),which is a key quantity of a model selection criterion, for linear regressionwith sparse regularization. Using the replica method, GDF is expressed by thevariables that characterize the saddle point of the free energy withoutdepending on the form of the regularization. Within the framework of replicasymmetric (RS) analysis, GDF is provided with a physical meaning as theeffective density of non-zero components. The validity of our method in the RSphase is supported by the consistency of our results with previous mathematicalresults. The analytical results in the RS phase are numerically achieved by thebelief propagation algorithm.

Evaluation of Generalized Degrees of Freedom for Sparse Estimation by Replica Method