Wavelet-based Bayesian approximate kernel method for high-dimensional data analysis

Kernel methods are often used for nonlinear regression and classification in statistics and machine learning because they are computationally cheap and accurate. The wavelet kernel functions based on wavelet analysis can efficiently approximate any nonlinear functions. In this article, we construct a novel wavelet kernel function in terms of random wavelet bases and define a linear vector space that captures nonlinear structures in reproducing kernel Hilbert spaces (RKHS). Based on the wavelet transform, the data are mapped into a low-dimensional randomized feature space and convert kernel function into operations of a linear machine. We then propose a new Bayesian approximate kernel model with the random wavelet expansion and use the Gibbs sampler to compute the model's parameters. Finally, some simulation studies and two real datasets analyses are carried out to demonstrate that the proposed method displays good stability, prediction performance compared to some other existing methods.