The periodic Sinc kernel: Theoretical design and applications in machine learning and scientific computing

Abstract

This paper proposes the data-dependent Sinc kernel function specifically designed for kernel-based machine learning tasks involving oscillatory and periodic data. Mercer’s theorem is proven for the proposed kernel, and its derivatives are explicitly computed. Notably, it is demonstrated that these derivatives form real symmetric positive definite Toeplitz matrices. To evaluate the effectiveness of the proposed kernel in machine learning and scientific applications, comprehensive assessments are conducted on a range of real-world and benchmark datasets, covering both periodic and non-periodic regression and classification tasks. Furthermore, the accuracy of the proposed kernel is validated through simulations involving different configurations of fractional Helmholtz, time-fractional sub-diffusion, and time-fractional Korteweg–de Vries differential equations on an unbounded domain. The results indicate that the proposed method outperforms existing periodic kernels, including Fourier and Wavelet kernels, in terms of accuracy. To facilitate the practical implementation and adoption of these findings, an open-source Python package named sinc is introduced at the end of this paper.