L_0 regularized estimation for nonlinear models that have sparse underlying linear structures

10/14/2009

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by   Zhiyi Chi, et al.

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We study the estimation of β for the nonlinear model y = f(Xβ) + ϵ when f is a nonlinear transformation that is known, β has sparse nonzero coordinates, and the number of observations can be much smaller than that of parameters (n≪ p). We show that in order to bound the L_2 error of the L_0 regularized estimator β̂, i.e., β̂ - β_2, it is sufficient to establish two conditions. Based on this, we obtain bounds of the L_2 error for (1) L_0 regularized maximum likelihood estimation (MLE) for exponential linear models and (2) L_0 regularized least square (LS) regression for the more general case where f is analytic. For the analytic case, we rely on power series expansion of f, which requires taking into account the singularities of f.