In this paper, we propose a variable selection method for general nonparametric kernel-based estimation. The proposed method consists of two-stage estimation: (1) construct a consistent estimator of the target function, (2) approximate the estimator using a few variables by l1-type penalized estimation. We see that the proposed method can be applied to various kernel nonparametric estimation such as kernel ridge regression, kernel-based density and density-ratio estimation. We prove that the proposed method has the property of the variable selection consistency when the power series kernel is used. This result is regarded as an extension of the variable selection consistency for the non-negative garrote to the kernel-based estimators. Several experiments including simulation studies and real data applications show the effectiveness of the proposed method.
Also, the non-negative garrote (NNG) was proposed by @cite_21 as a modification of the standard least square (LS) estimator for linear models. In the NNG, each coefficient of the LS estimator is shrunk towards zero, and its intensity is controlled by the @math -regularization. The variable selection consistency of Lasso was proved under the so-called irrepresentable condition @cite_25 @cite_23 . On the other hand, @cite_13 and @cite_22 proved that the NNG has the variable selection consistency without the irrepresentable condition. For some kernel-based estimators, @cite_18 and @cite_4 employed the NNG as the adaptive scaling for the variable selection. Also, @cite_6 proved the variable selection consistency of the scaled kernel-ridge estimator under a variant of the irrepresentable condition.
@cite_18 @cite_4 @cite_22 @cite_21 @cite_6 @cite_23 @cite_13 @cite_25
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A new method, called the nonnegative (nn) garrote, is proposed for doing subset regression. It both shrinks and zeroes coefficients. In tests on real and simulated data, it produces lower prediction error than ordinary subset selection. It is also compared to ridge regression. If the regression equations generated by a procedure do not change drastically with small changes in the data, the procedure is called stable. Subset selection is unstable, ridge is very stable, and the nn-garrote is intermediate. Simulation results illustrate the effects of instability on prediction error. Summary Variable selection for recovering sparsity in nonadditive and nonparametric models with high-dimensional variables has been challenging. This problem becomes even more difficult due to complications in modeling unknown interaction terms among high-dimensional variables. There is currently no variable selection method to overcome these limitations. Hence, in this article we propose a variable selection approach that is developed by connecting a kernel machine with the nonparametric regression model. The advantages of our approach are that it can: (i) recover the sparsity; (ii) automatically model unknown and complicated interactions; (iii) connect with several existing approaches including linear nonnegative garrote and multiple kernel learning; and (iv) provide flexibility for both additive and nonadditive nonparametric models. Our approach can be viewed as a nonlinear version of a nonnegative garrote method. We model the smoothing function by a Least Squares Kernel Machine (LSKM) and construct the nonnegative garrote objective function as the function of the sparse scale parameters of kernel machine to recover sparsity of input variables whose relevances to the response are measured by the scale parameters. We also provide the asymptotic properties of our approach. We show that sparsistency is satisfied with consistent initial kernel function coefficients under certain conditions. An efficient coordinate descent backfitting algorithm is developed. A resampling procedure for our variable selection methodology is also proposed to improve the power. Summary. We study the non-negative garrotte estimator from three different aspects: consistency, computation and flexibility. We argue that the non-negative garrotte is a general procedure that can be used in combination with estimators other than the original least squares estimator as in its original form. In particular, we consider using the lasso, the elastic net and ridge regression along with ordinary least squares as the initial estimate in the non-negative garrotte. We prove that the non-negative garrotte has the nice property that, with probability tending to 1, the solution path contains an estimate that correctly identifies the set of important variables and is consistent for the coefficients of the important variables, whereas such a property may not be valid for the initial estimators. In general, we show that the non-negative garrotte can turn a consistent estimate into an estimate that is not only consistent in terms of estimation but also in terms of variable selection. We also show that the non-negative garrotte has a piecewise linear solution path. Using this fact, we propose an efficient algorithm for computing the whole solution path for the non-negative garrotte. Simulations and a real example demonstrate that the non-negative garrotte is very effective in improving on the initial estimator in terms of variable selection and estimation accuracy. Meinshausen and Buhlmann [Ann. Statist. 34 (2006) 1436-1462] showed that, for neighborhood selection in Gaussian graphical models, under a neighborhood stability condition, the LASSO is consistent, even when the number of variables is of greater order than the sample size. Zhao and Yu [(2006) J. Machine Learning Research 7 2541-2567] formalized the neighborhood stability condition in the context of linear regression as a strong irrepresentable condition. That paper showed that under this condition, the LASSO selects exactly the set of nonzero regression coefficients, provided that these coefficients are bounded away from zero at a certain rate. In this paper, the regression coefficients outside an ideal model are assumed to be small, but not necessarily zero. Under a sparse Riesz condition on the correlation of design variables, we prove that the LASSO selects a model of the correct order of dimensionality, controls the bias of the selected model at a level determined by the contributions of small regression coefficients and threshold bias, and selects all coefficients of greater order than the bias of the selected model. Moreover, as a consequence of this rate consistency of the LASSO in model selection, it is proved that the sum of error squares for the mean response and the l α -loss for the regression coefficients converge at the best possible rates under the given conditions. An interesting aspect of our results is that the logarithm of the number of variables can be of the same order as the sample size for certain random dependent designs.