Variable transformations in combination with wavelets and ANOVA for high-dimensional approximation

We use hyperbolic wavelet regression for the fast reconstruction of high-dimensional functions having only low-dimensional variable interactions. Compactly supported periodic Chui-Wang wavelets are used for the tensorized hyperbolic wavelet basis on the torus. With a variable transformation, we are...

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Vydáno v:Advances in computational mathematics Ročník 50; číslo 3; s. 53
Hlavní autoři: Potts, Daniel, Weidensager, Laura
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.06.2024
Springer Nature B.V
Témata:
Approximation
Computational Mathematics and Numerical Analysis
Computational Science and Engineering
Density functions
Least squares method
Mathematical and Computational Biology
Mathematical Modeling and Industrial Mathematics
Mathematics
Mathematics and Statistics
Periodic functions
Polynomials
Statistical analysis
Toruses
Transformations (mathematics)
Variance analysis
Visualization
Wavelets
ANOVA decomposition
41A25
Random sampling
41A63
Variable transformations
65D15
Least squares approximation
65T60
ISSN:1019-7168, 1572-9044
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Popis
Shrnutí:We use hyperbolic wavelet regression for the fast reconstruction of high-dimensional functions having only low-dimensional variable interactions. Compactly supported periodic Chui-Wang wavelets are used for the tensorized hyperbolic wavelet basis on the torus. With a variable transformation, we are able to transform the approximation rates and fast algorithms from the torus to other domains. We perform and analyze scattered data approximation for smooth but arbitrary density functions by using a least squares method. The corresponding system matrix is sparse due to the compact support of the wavelets, which leads to a significant acceleration of the matrix vector multiplication. For non-periodic functions, we propose a new extension method. A proper choice of the extension parameter together with the piecewise polynomial Chui-Wang wavelets extends the functions appropriately. In every case, we are able to bound the approximation error with high probability. Additionally, if the function has a low effective dimension (i.e., only interactions of a few variables), we qualitatively determine the variable interactions and omit ANOVA terms with low variance in a second step in order to decrease the approximation error. This allows us to suggest an adapted model for the approximation. Numerical results show the efficiency of the proposed method.
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ISSN:1019-7168
1572-9044
DOI:10.1007/s10444-024-10147-2