Least Square Regression with Non-Identical Unbounded Sampling and Coefficient Regularization

Xinxin Chang, Lu Luo

Abstract: In this paper, we consider a coefficient-based least squares regression problem with indefinite kernels from non-identical unbounded sampling processes. Here non-identical unbounded sampling means the samples are drawn independently but not identically from unbounded sampling processes. And except for continuity and boundedness, the kernel function is not necessary to satisfy any further regularity conditions. This lead to additional difficulty. By introducing a suitable reproducing kernel Hilbert space (RKHS) and a suitable intermediate integral operator, and by the error decomposition procedure the sample error is divided into two parts. we deduce the error bound. Last we yield satisfactory results by proper choice of the regularization parameter.

Keywords: Indefinite Kernel, Coefficient Regularization, Least Square Regression, Integral Operator, Unbounded Hypothesis, Learning Rates