A New Interpolation Approach and Corresponding Instance-Based Learning

TL;DR

This paper introduces ADB interpolation, a novel high-dimensional interpolation method that enhances instance-based learning with advantages like simplicity, stability, and interpretability, suitable for big data applications.

Contribution

It proposes a new interpolation approach called ADB interpolation and develops a corresponding instance-based learning method with theoretical and practical benefits.

Findings

ADB interpolation achieves stable accuracy in high-dimensional spaces.

The learning method is efficient, interpretable, and avoids misclassification.

Applicable to vector-valued functions and complements deep learning.

Abstract

Starting from finding approximate value of a function, introduces the measure of approximation-degree between two numerical values, proposes the concepts of "strict approximation" and "strict approximation region", then, derives the corresponding one-dimensional interpolation methods and formulas, and then presents a calculation model called "sum-times-difference formula" for high-dimensional interpolation, thus develops a new interpolation approach, that is, ADB interpolation. ADB interpolation is applied to the interpolation of actual functions with satisfactory results. Viewed from principle and effect, the interpolation approach is of novel idea, and has the advantages of simple calculation, stable accuracy, facilitating parallel processing, very suiting for high-dimensional interpolation, and easy to be extended to the interpolation of vector valued functions. Applying the approach to instance-based learning, a new instance-based learning method, learning using ADB interpolation, is obtained. The learning method is of unique technique, which has also the advantages of definite mathematical basis, implicit distance weights, avoiding misclassification, high efficiency, and wide range of applications, as well as being interpretable, etc. In principle, this method is a kind of learning by analogy, which and the deep learning that belongs to inductive learning can complement each other, and for some problems, the two can even have an effect of "different approaches but equal results" in big data and cloud computing environment. Thus, the learning using ADB interpolation can also be regarded as a kind of "wide learning" that is dual to deep learning.