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Nonlinear function estimation is core to modern machine learningapplications. In this paper, to perform nonlinear function estimation, wereduce a nonlinear inverse problem to a linear one using a polynomial kernelexpansion. These kernels increase the feature set, and may result in poorlyconditioned matrices. Nonetheless, we show several examples where the matrix inour linear inverse problem contains only mild linear correlations amongcolumns. The coefficients vector is modeled within a Bayesian setting for whichapproximate message passing (AMP), an algorithmic framework for signalreconstruction, offers Bayes-optimal signal reconstruction quality. While theBayesian setting limits the scope of our work, it is a first step towardestimation of real world nonlinear functions. The coefficients vector isestimated using two AMP-based approaches, a Bayesian one and empirical Bayes.Numerical results confirm that our AMP-based approaches learn the functionbetter than LASSO, offering markedly lower error in predicting test data.

Nonlinear Function Estimation with Empirical Bayes and Approximate Message Passing