Samplets: A new paradigm for data compression

TL;DR

Samplets introduce a multilevel data representation technique that enables efficient data compression, singularity detection, and matrix approximation, significantly improving the handling of large datasets in kernel-based learning.

Contribution

This paper presents the novel concept of samplets, extending wavelet ideas to data representation for compression and analysis, with practical algorithms for kernel matrix approximation.

Findings

Samplets achieve O(N log N) matrix compression.

Thresholding small entries yields quasi-sparse matrices.

Numerical studies confirm efficiency and effectiveness.

Abstract

In this article, we introduce the concept of samplets by transferring the construction of Tausch-White wavelets to the realm of data. This way we obtain a multilevel representation of discrete data which directly enables data compression, detection of singularities and adaptivity. Applying samplets to represent kernel matrices, as they arise in kernel based learning or Gaussian process regression, we end up with quasi-sparse matrices. By thresholding small entries, these matrices are compressible to O(N log N) relevant entries, where N is the number of data points. This feature allows for the use of fill-in reducing reorderings to obtain a sparse factorization of the compressed matrices. Besides the comprehensive introduction to samplets and their properties, we present extensive numerical studies to benchmark the approach. Our results demonstrate that samplets mark a considerable step in the direction of making large data sets accessible for analysis.