Approximation errors of online sparsification criteria

TL;DR

This paper develops a theoretical framework to analyze and bound the approximation errors of various online sparsification criteria used in machine learning models, improving understanding of their efficiency and accuracy.

Contribution

It introduces bounds on approximation errors for multiple sparsification criteria, linking them to feature approximation in online learning models.

Findings

Derived theoretical bounds on approximation errors for sparsification criteria

Connected sparsification criteria to feature approximation errors

Provided upper bounds for empirical mean and kernel PCA features

Abstract

Many machine learning frameworks, such as resource-allocating networks, kernel-based methods, Gaussian processes, and radial-basis-function networks, require a sparsification scheme in order to address the online learning paradigm. For this purpose, several online sparsification criteria have been proposed to restrict the model definition on a subset of samples. The most known criterion is the (linear) approximation criterion, which discards any sample that can be well represented by the already contributing samples, an operation with excessive computational complexity. Several computationally efficient sparsification criteria have been introduced in the literature, such as the distance, the coherence and the Babel criteria. In this paper, we provide a framework that connects these sparsification criteria to the issue of approximating samples, by deriving theoretical bounds on the approximation errors. Moreover, we investigate the error of approximating any feature, by proposing upper-bounds on the approximation error for each of the aforementioned sparsification criteria. Two classes of features are described in detail, the empirical mean and the principal axes in the kernel principal component analysis.