Sensitivity Analysis on the Sphere and a Spherical ANOVA Decomposition

Technical Summary: Sensitivity Analysis on the Sphere and a Spherical ANOVA Decomposition

Problem Statement
The Analysis of Variance (ANOVA) decomposition is a standard framework for understanding the structure of multivariate functions, particularly for sensitivity analysis and dimension reduction. In classical settings where the domain is a tensor product space (e.g., $[0,1]^d$), the decomposition relies on a natural product structure that ensures the orthogonality of component functions corresponding to different variable subsets. However, extending this framework to the high-dimensional sphere $S^d$ presents significant theoretical challenges. The sphere lacks a tensor product structure, and the constraint $\|x\|=1$ induces intrinsic dependencies between input variables. Consequently, classical definitions of ANOVA components, which rely on integrating over independent variables, fail to produce unique or orthogonal terms on the sphere. Furthermore, existing generalizations for dependent variables often require grid-closed domains or specific coordinate transformations that compromise the interpretability of the original physical variables.

Methodology
The paper proposes a novel spherical ANOVA decomposition tailored for functions $f: S^d \to \mathbb{R}$ under the uniform distribution. The methodology addresses the geometric constraints of the sphere through three primary mechanisms:

1. Parity Decomposition: Unlike the classical case, the authors introduce a parity vector $\xi \in \{0, 1\}^{d+1}$ to decompose terms into even and odd parts. This is necessary because the algebraic dependence of variables on the sphere ($x_i^2$ terms) creates redundancies that cannot be resolved without distinguishing parity. The decomposition takes the form:
   $$f(x) = \sum_{u \in \mathcal{D}} \sum_{\xi \in \Upsilon_u} f_{u,\xi}(x_u)$$
   where $u$ is a subset of indices and $\xi$ indicates the parity of the term with respect to each variable.

2. Iterative Projection Operators: The authors define a projection operator $P_u$ that integrates a function over the "fiber" of the sphere (a lower-dimensional sphere) corresponding to fixed values of the variables in $u$. Unlike the classical tensor product projection, this operator does not factorize. The ANOVA terms are then defined iteratively using these projections, similar to the classical Hoeffding decomposition but adapted for the spherical geometry.

3. Redundancy Elimination: Due to the relation $\sum x_i^2 = 1$, even terms depending on a variable $x_{d+1}$ can be expressed as functions of the remaining variables. To ensure a unique and non-redundant decomposition, the authors propose omitting specific terms where the highest index variable appears with even parity. This reduces the total number of terms and prevents linear dependencies in the basis.

4. Orthogonal Basis Construction: To implement the decomposition numerically, the paper constructs reduced orthogonal bases for the lower-dimensional balls $B^{|u|}$ on which the terms $f_{u,\xi}$ are defined. These bases utilize Gegenbauer polynomials for one-dimensional terms and Jacobi polynomials combined with spherical harmonics for higher-dimensional terms, ensuring orthogonality with respect to the induced weight functions on the sphere.

5. Numerical Approximation and Sensitivity Analysis: The decomposition is approximated using a truncated series of these basis functions. The coefficients are determined by solving a least-squares problem (using the LSQR algorithm) on random sampling nodes. Based on the variances of these terms, the authors define spherical Sobol indices to quantify variable importance, acknowledging that while strict orthogonality is lost for even terms, the resulting indices provide stable measures of variable influence.

Key Contributions

• Spherical ANOVA Definition: The paper establishes the first dedicated ANOVA decomposition for the sphere that retains the interpretability of original variables, avoiding the need for coordinate system transformations (e.g., spherical coordinates) that obscure physical meaning.
• Parity-Based Structure: It introduces the necessity of distinguishing between even and odd terms via a parity vector $\xi$, a feature absent in classical ANOVA but essential for handling the sphere's geometry.
• Theoretical Properties: The authors analyze the transferability of classical properties, demonstrating that while orthogonality holds between terms of different parity, it does not strictly hold between terms of the same parity due to variable dependencies. They provide specific integral conditions that the terms must satisfy.
• Reduced Basis and Redundancy Handling: The work defines a method to identify and omit redundant terms arising from the sphere's constraint, ensuring a linearly independent dictionary of basis functions.
• Sensitivity Indices: The paper defines Sobol indices adapted for the spherical domain, providing a framework for global sensitivity analysis on dependent inputs constrained to a manifold.

Results
Numerical experiments were conducted on the 10-dimensional sphere ($S^{10}$) using test functions with known interaction structures (including polynomials, mixed parity functions, and the Ishigami function).

• The proposed method successfully identified non-zero ANOVA terms corresponding to the known interactions in the test functions.
• For functions composed entirely of odd terms, the decomposition closely matched expectations, with non-zero terms detected accurately.
• For functions involving even terms, the method correctly identified the primary interacting variables, though some variance was distributed to higher-order terms due to the inherent dependencies. The authors note that the magnitude of these "misleading" terms is small compared to the primary terms.
• The iterative approach of increasing the order of terms (order 1, then order 2) yielded results comparable to solving for all terms simultaneously, validating the stability of the LSQR solver in this context.

Significance and Claims
The paper claims that the proposed spherical ANOVA decomposition provides a "meaningful and interpretable representation" of functions on the sphere, filling a gap in the literature where no dedicated decomposition existed. Its significance lies in:

• Geometry-Aware Analysis: It respects the intrinsic curvature and variable coupling of the sphere, which classical methods fail to capture without losing interpretability.
• Dimension Reduction: By identifying that high-dimensional functions on the sphere are often dominated by low-order interactions, the method offers a pathway to mitigate the curse of dimensionality in approximation and integration.
• Sensitivity Analysis: It enables the quantification of variable importance in settings with dependent inputs, serving as a tool for feature selection and noise reduction in fields dealing with normalized feature vectors (e.g., geophysics, quantum mechanics).

The authors maintain a modest tone, acknowledging that the decomposition does not answer all application-based questions (such as specific function space characterizations) and that the lack of strict orthogonality for even terms means the Sobol indices should be interpreted as relative importance measures rather than exact variance partitions. The work is presented as a foundational theoretical framework and a practical algorithm for analyzing functions on spherical domains.