# Samplets: Construction and Scattered Data Compression

26 Pages Posted: 22 Mar 2022

See all articles by Helmut Harbrecht

## Helmut Harbrecht

affiliation not provided to SSRN

## Michael D. Multerer

Swiss Finance Institute - USI Lugano

### Abstract

We introduce the concept of samplets by transferring the construction of Tausch-White wavelets to scattered data. This way, we obtain a multiresolution analysis tailored to discrete data which directly enables data compression, feature detection and adaptivity. The cost for constructing the samplet basis and for the fast samplet transform, respectively,is $$\Ocal(N)$$, where $$N$$ is the number of data points.Samplets with vanishing moments can be used to compresskernel matrices, arising, for instance, in kernel based learning or Gaussian process regression. This leads tosparse matrices with only $$\mathcal{O}(N\log N)$$remaining entries. We provide estimates for the compression error and present an algorithm that computes the compressed kernel matrix with computationalcost $$\mathcal{O}(N\log N)$$. The accuracy of the approximation is controlled by the number of vanishing moments.Besides the cost efficient storage of kernel matrices,the sparse representation enables the application of sparsedirect solvers for the numerical solution of corresponding linear systems.In addition to a comprehensive introduction to samplets andtheir properties, we present numerical studies to benchmarkthe approach. Our results demonstrate that samplets mark aconsiderable step in the direction of making large scattered datasets accessible for multiresolution analysis.

Suggested Citation

Harbrecht, Helmut and Multerer, Michael D., Samplets: Construction and Scattered Data Compression. Available at SSRN: https://ssrn.com/abstract=4053305 or http://dx.doi.org/10.2139/ssrn.4053305