Analyzing sparse dictionaries for online learning with kernels

TL;DR

This paper analyzes various sparsity measures for dictionaries in online kernel learning, demonstrating their shared properties and establishing a quasi-isometry between parameter space and feature space.

Contribution

It provides a unified eigenvalue analysis of sparsity measures, revealing their common properties and the relationship between parameter and feature spaces in online kernel learning.

Findings

Sparsity measures share properties like linear independence and well-posedness.

Eigenvalue analysis links sparsity measures to dictionary properties.

Existence of a quasi-isometry between parameter and feature spaces.

Abstract

Many signal processing and machine learning methods share essentially the same linear-in-the-parameter model, with as many parameters as available samples as in kernel-based machines. Sparse approximation is essential in many disciplines, with new challenges emerging in online learning with kernels. To this end, several sparsity measures have been proposed in the literature to quantify sparse dictionaries and constructing relevant ones, the most prolific ones being the distance, the approximation, the coherence and the Babel measures. In this paper, we analyze sparse dictionaries based on these measures. By conducting an eigenvalue analysis, we show that these sparsity measures share many properties, including the linear independence condition and inducing a well-posed optimization problem. Furthermore, we prove that there exists a quasi-isometry between the parameter (i.e., dual) space and the dictionary's induced feature space.