Dimension Independent Data Sets Approximation and Applications to Classification

TL;DR

This paper introduces a kernel-based approximation method for discrete data sets that produces smooth functions for robust classification, demonstrated on low-dimensional examples and the high-dimensional MNIST dataset.

Contribution

It presents a novel kernel approximation approach that is dimension-independent and robust for classification tasks, especially effective on high-dimensional data.

Findings

Successful classification on low-dimensional examples.

Effective application to high-dimensional MNIST dataset.

Robustness of the method to data variations.

Abstract

We revisit the classical kernel method of approximation/interpolation theory in a very specific context motivated by the desire to obtain a robust procedure to approximate discrete data sets by (super)level sets of functions that are merely continuous at the data set arguments but are otherwise smooth. Special functions, called data signals, are defined for any given data set and are used to succesfully solve supervised classification problems in a robust way that depends continuously on the data set. The efficacy of the method is illustrated with a series of low dimensional examples and by its application to the standard benchmark high dimensional problem of MNIST digit classification.