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Nowadays, several data analysis problems require for complexity reduction,mainly meaning that they target at removing the non-influential covariates fromthe model and at delivering a sparse model. When categorical covariates arepresent, with their levels being dummy coded, the number of parameters includedin the model grows rapidly, fact that emphasizes the need for reducing thenumber of parameters to be estimated. In this case, beyond variable selection,sparsity is also achieved through fusion of levels of covariates which do notdifferentiate significantly in terms of their influence on the responsevariable. In this work a new regularization technique is introduced, called$L_{0}$-Fused Group Lasso ($L_{0}$-FGL) for binary logistic regression. It usesa group lasso penalty for factor selection and for the fusion part it appliesan $L_{0}$ penalty on the differences among the levels' parameters of acategorical predictor. Using adaptive weights, the adaptive version of$L_{0}$-FGL method is derived. Theoretical properties, such as the existence,$\sqrt{n}$ consistency and oracle properties under certain conditions, areestablished. In addition, it is shown that even in the diverging case where thenumber of parameters $p_{n}$ grows with the sample size $n$, $\sqrt{n}$consistency and a consistency in variable selection result are achieved. Twocomputational methods, PIRLS and a block coordinate descent (BCD) approachusing quasi Newton, are developed and implemented. A simulation study supportsthat $L_{0}$-FGL shows an outstanding performance, especially in the highdimensional case.

Simultaneous Factors Selection and Fusion of Their Levels in Penalized Logistic Regression