Abstract
We consider the moving least-squares (MLS) method by the regression learning framework under the assumption that the sampling process satisfies the α-mixing condition. We conduct the rigorous error analysis by using the probability inequalities for the dependent samples in the error estimates. When the dependent samples satisfy an exponential α-mixing, we derive the satisfactory learning rate and error bound of the algorithm.
Highlights
The least-squares (LS) method is an important global approximate method based on the regular or concentrated data sample points
The moving least-squares (MLS) method was introduced by McLain in [4] to draw a set of contours based on a cluster of scattered data sample points
The goal of regression learning is to find a good approximation of the regression function fρ based on a set of random samples z = {zi}mi=1 = {(xi, yi)}mi=1 ∈ Zm drawn according to the measure ρ
Summary
The least-squares (LS) method is an important global approximate method based on the regular or concentrated data sample points. There are still some irregular or scattered samples which are obtained in many practical applications such as engineering and machine learning [1,2,3,4] They need to be analyzed to achieve their special usefulness. The goal of regression learning is to find a good approximation of the regression function fρ based on a set of random samples z = {zi}mi=1 = {(xi, yi)}mi=1 ∈ Zm drawn according to the measure ρ. The proof is analogous to that of Theorem 3 in [8] except that we need to use the following Lemma 3.1 for the dependent sampling setting to replace Lemma 2 in [8]. If (1.7) and (2.1) hold, with confidence 1 – δ, we have
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