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Samplets: Construction and scattered data compression
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DOI：10.1016/j.jcp.2022.111616.png)
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Abstract
En 中文
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 dis-crete 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 O(N), where N is the number of data points. Samplets with vanishing moments can be used to compress kernel matrices, arising, for instance, in kernel based learning and scattered data approximation. The result are sparse matrices with only O(N log N) remain-ing entries. We provide estimates for the compression error and present an algorithm that computes the compressed kernel matrix with computational cost O(N log N). The accu-racy of the approximation is controlled by the number of vanishing moments. Besides the cost efficient storage of kernel matrices, the sparse representation enables the applica-tion of sparse direct solvers for the numerical solution of corresponding linear systems. In addition to a comprehensive introduction to samplets and their properties, we present numerical studies to benchmark the approach. Our results demonstrate that samplets mark a considerable step in the direction of making large scattered data sets accessible for mul-tiresolution analysis.(c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Keywords:
Multiresolution analysis
Unstructured data
Kernel methods
Data compression
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