ANOVA approximation with mixed tensor product basis on scattered points

TL;DR

The paper develops an ANOVA-type decomposition for high-dimensional functions using a mixed tensor-product basis built from Fourier, half-period cosine, and Chebyshev functions, enabling interpretable sensitivity analysis on scattered data. A fast evaluation framework, including non-equidistant fast transforms (NFMT) and grouped transformations, makes computing mixed-polynomial approximations scalable via LSQR and carefully chosen truncation sets U guided by analytic global sensitivity indices. The approach yields accurate, interpretable models that adapt the basis per dimension and interaction order, with numerical experiments showing clear advantages over single-basis alternatives and real-data improvements in predictive error and interpretability. Overall, the work provides a practical, scalable toolkit for mixed-basis ANOVA approximations and sensitivity analysis on nonuniformly sampled high-dimensional data.

Abstract

In this paper we consider an orthonormal basis, generated by a tensor product of Fourier basis functions, half period cosine basis functions, and the Chebyshev basis functions. We deal with the approximation problem in high dimensions related to this basis and design a fast algorithm to multiply with the underlying matrix, consisting of rows of the non-equidistant Fourier matrix, the non-equidistant cosine matrix and the non-equidistant Chebyshev matrix, and its transposed. This leads us to an ANOVA (analysis of variance) decomposition for functions with partially periodic boundary conditions through using the Fourier basis in some dimensions and the half period cosine basis or the Chebyshev basis in others. We consider sensitivity analysis in this setting, in order to find an adapted basis for the underlying approximation problem. More precisely, we find the underlying index set of the multidimensional series expansion. Additionally, we test this ANOVA approximation with mixed basis at numerical experiments, and refer to the advantage of interpretable results.

Figures (8)

• Figure 1: Frequency set $\mathcal{I}(U)$ for $\mathbf{d}=\left(\exp\mathop{\mathrm{alg}}\cos\right)$ and $U=\{\emptyset,\{1\},\{2\},\{3\},\{1,2\},\{2,3\}\}$ with $\mathbf{N}^{\{1\}}=\left(1800\right)$, $\mathbf{N}^{\{2\}}=\left(0160\right)$, $\mathbf{N}^{\{3\}}=\left(0010\right)$, $\mathbf{N}^{\{1,2\}}=\left(1080\right)$, and $\mathbf{N}^{\{2,3\}}=\left(068\right)$. The frequency set $\tilde{\mathcal{I}}_{\mathbf{N}^{\{1\}}}^\mathbf{d}$ is shown in green, $\tilde{\mathcal{I}}_{\mathbf{N}^{\{2\}}}^\mathbf{d}$ in blue, $\tilde{\mathcal{I}}_{\mathbf{N}^{\{3\}}}^\mathbf{d}$ in red, $\tilde{\mathcal{I}}_{\mathbf{N}^{\{1,2\}}}^\mathbf{d}$ in cyan, and $\tilde{\mathcal{I}}_{\mathbf{N}^{\{2,3\}}}^\mathbf{d}$ in magenta.
• Figure 2: Mean squared errors for $U_2$ and bandwidths $\mathbf{N}^\mathbf{u}=(\lvert{\{i\}\cap\mathbf{u}}\rvert N_{\lvert{\mathbf{u}}\rvert})_{i=1}^4$ and $N_1$ and $N_2$ for $f_1$.
• Figure 3: Approximated global sensitivity indices for $U_2$ and bandwidths $\mathbf{N}^\mathbf{u}=(\lvert{\{i\}\cap\mathbf{u}}\rvert N_{\lvert{\mathbf{u}}\rvert})_{i=1}^4$ and $N_1=12$ and $N_2=10$ for $f_1$.
• Figure 4: Boxplot of the MSEs for the approximation of $f_1$ with mixed and cosine basis.
• Figure 5: Mean squared errors of the ANOVA approximation $\tilde{f}_2$ to $f_2$ for the bandwidths from Table \ref{['tb:3']} and $M$ training nodes, evaluated at 10000 nodes.

Theorems & Definitions (11)

• Lemma 2.1
• proof
• Remark
• Definition 2.2
• Lemma 3.1
• proof
• Example
• Theorem 4.1
• proof
• Remark