Coefficient-Based Regression with Non-Identical Unbounded Sampling

Jia Cai

doi:10.1155/2013/134727

Abstract

We investigate 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. The kernel is not necessarily symmetric or positive semi-definite. This leads to additional difficulty in the error analysis. By introducing a suitable reproducing kernel Hilbert space (RKHS) and a suitable intermediate integral operator, elaborate analysis is presented by means of a novel technique for the sample error. This leads to satisfactory results.

Highlights

Introduction and PreliminaryWe study coefficient-based least squares regression with indefinite kernels from non-identical unbounded sampling processes
We investigate a coefficient-based least squares regression problem with indefinite kernels from non-identical unbounded sampling processes
We consider the indefinite kernel scheme in a hypothesis space HK,z depending on the sample z; this space is defined by HK,z = {∑βiKxi : βi ∈ R}

Summary

Introduction and Preliminary

We study coefficient-based least squares regression with indefinite kernels from non-identical unbounded sampling processes. Holder ≤ 1) is defined as the space of all continuous functions on X with the following norm finite [1]:. Classical learning algorithm is conducted by a scheme in a reproducing kernel Hilbert space (RKHS) [2] associated with a Mercer kernel K : X × X → R, which is defined to be a continuous, symmetric, and positive semi-definite (p.s.d.). We consider the indefinite kernel scheme in a hypothesis space HK,z depending on the sample z; this space is defined by HK,z = {∑βiKxi : βi ∈ R}. We study learning algorithm (8) by non-identical unbounded sampling processes with indefinite kernels. The major contribution we make is on the sample error estimate; the main difficulty is the non-identical unbounded sampling of the samples; we overcome this difficulty by introducing a suitable intermediate operator

Key Analysis and Main Results

Error Analysis