Sensitivity Analysis on the Sphere and a Spherical ANOVA Decomposition

Laura Weidensager

arxiv: 2512.23476 · v2 · pith:XTHE4LPDnew · submitted 2025-12-29 · 🧮 math.NA · cs.NA

Sensitivity Analysis on the Sphere and a Spherical ANOVA Decomposition

Laura Weidensager This is my paper

Pith reviewed 2026-05-16 19:17 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords sensitivity analysisANOVA decompositionspherical functionsvariable interactionfunction approximationorthogonal basisparity decomposition

0 comments

The pith

A function on the sphere decomposes into a sum of terms each depending on a coordinate subset and a parity vector.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to break down any real-valued function defined on the d-dimensional sphere into additive pieces. Each piece depends only on a small number of the input coordinates together with an extra index that tracks even or odd behavior. This extends the classical ANOVA decomposition by using the sphere's geometry to group the variables naturally. A reader would care because it opens the door to sensitivity analysis and approximation of high-dimensional spherical functions using only low-order interactions.

Core claim

We establish sensitivity analysis on the sphere by presenting formulas that decompose a function f from the d-sphere to the reals into a sum of terms f_{u,ξ} indexed by subsets u of the d+1 coordinates and a parity vector ξ. Each such term depends only on the variables in u, and the decomposition is enabled by orthogonal basis functions suited to the sphere.

What carries the argument

The spherical ANOVA decomposition with parity, which splits functions using subsets of coordinates and even/odd parity to capture geometry-induced dependencies.

Load-bearing premise

The natural geometry on the sphere leads to specific dependencies between the input variables for which suitable orthogonal basis functions permit the decomposition and approximation.

What would settle it

A calculation showing that the proposed decomposition formula does not sum back to the original function for some test case on the sphere would disprove the claim.

Figures

Figures reproduced from arXiv: 2512.23476 by Laura Weidensager.

**Figure 1.** Figure 1: Visualization of P{1}f for d = 2: Integration over the red circle with fixed x1 = y1. Applying the operator Pa to a function, which depends only on the variables in u leads to the following lemma. This will be helpful in Section 3.2 to define integral conditions for the ANOVA operator. Lemma 3.4. If u ∈ D\∅ and for a ∈ D with a ⊂ u, the operator Pa applied to a function gu : B |u| → R, which is naturally a… view at source ↗

**Figure 2.** Figure 2: Visualization of all possible indices u and ξ of the terms fu,ξ for the case d = 3. As calculated in (3.2), we have in total 49 terms. In Definition 3.9 we defined these terms iteratively. We illustrate this definition by using different colors for the terms belonging to different parity vectors ξ. In Section 3.5 we explain which terms have to be omitted due to redundancies on the sphere. We have grayed ou… view at source ↗

**Figure 3.** Figure 3: Illustration of the indices u, v, b ⊆ [d + 1]. Lemma 3.12. Let the function f(x) have the form f(x) = g(xu), depend on all variables in u and have parity ξ, i.e. Ξξf = f. Let furthermore the parity vector ξ have the form ξ = (1v, 0vc ) ⊤ with v ⊂ u. Then for all ∅ ̸= b ⊆ [d + 1]\u we have for the spherical ANOVA terms defined in Definition 3.9 that 0 = X v⊆a⊆u fa∪b,ξ. (3.12) Proof. We illustrate the indice… view at source ↗

**Figure 4.** Figure 4: Approximated Sobol indices (5.3) for the different test functions. We plot the results from LSQR with all [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗

read the original abstract

We establish sensitivity analysis on the sphere. We present formulas that allow us to decompose a function $f\colon \mathbb S^d\rightarrow \mathbb R$ into a sum of terms $f_{\boldsymbol u,\boldsymbol \xi}$. The index $\boldsymbol u$ is a subset of $\{1,2,\ldots,d+1\}$, where each term $f_{\boldsymbol u,\boldsymbol \xi}$ depends only on the variables with indices in $\boldsymbol u$. In contrast to the classical analysis of variance (ANOVA) decomposition, we additionally use the decomposition of a function into functions with different parity, which adds the additional parameter $\boldsymbol \xi$. The natural geometry on the sphere naturally leads to the dependencies between the input variables. Using certain orthogonal basis functions for the function approximation, we are able to model high-dimensional functions with low-dimensional variable interactions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives formulas for a parity-augmented ANOVA decomposition on the sphere that respects the geometry, but the completeness of the sum under the spherical measure needs explicit verification.

read the letter

The main takeaway is that this paper adapts the classical ANOVA decomposition to functions on the sphere by adding a parity index ξ alongside the usual subset index u. The result is a way to break f into terms each depending on a small number of coordinates while accounting for the sphere's quadratic constraint. That extension is the actual novelty here, and it lines up with needs in geophysics or manifold optimization where variables are not independent in the usual Euclidean sense. The paper does well by noting that the sphere's geometry itself induces the variable dependencies and by pointing to orthogonal bases that let high-dimensional functions be approximated via low-order interactions. That framing is practical and avoids forcing a flat-space decomposition onto a curved domain. The soft spot is the completeness question raised in the stress-test note. The abstract asserts that f equals the sum of the f_u,ξ terms, but the sphere couples all coordinates globally, so any basis must be shown to produce orthogonal projections whose sum recovers f exactly in L² under the spherical measure. Without an explicit construction (such as symmetrized spherical harmonics) and a short proof that cross terms vanish and nothing is missed, the decomposition could be incomplete or non-unique. The abstract does not display that step, so the central claim rests on an unshown argument. This work is aimed at readers already comfortable with Sobol-style indices and spherical harmonics who want to extend sensitivity tools to spherical domains. A numerical analyst or uncertainty-quantification person working on climate or optimization problems would get direct value from the formulas. It deserves a serious referee because the idea is coherent, the motivation is clear, and the gap is fixable with one added section rather than a rewrite. I would send it out for review.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to establish sensitivity analysis on the sphere by providing formulas to decompose a function f : S^d → R into a sum of terms f_{u,ξ}, where u is a subset of {1,2,…,d+1} and ξ encodes parity. This extends classical ANOVA by incorporating parity decomposition, using orthogonal basis functions to model high-dimensional functions with low-dimensional variable interactions induced by the sphere's geometry.

Significance. If the central decomposition holds with complete orthogonal projections, the result would offer a novel framework for sensitivity analysis in spherical settings, potentially useful for high-dimensional approximation where variable dependencies arise naturally from the quadratic constraint. The use of parity adds an interesting dimension to the decomposition.

major comments (2)

[Abstract and §2] The completeness of the decomposition, i.e., that the sum of all f_{u,ξ} recovers f exactly under the spherical measure, is asserted but requires an explicit proof that the chosen orthogonal basis induces projections that are complete in L²(S^d) and that no cross terms arise from the coordinate coupling; without this, the claim that each term depends only on variables in u may not hold uniquely.
[§4] The specific orthogonal basis functions are referred to but not constructed in detail; an example or definition (such as restrictions of multivariate polynomials or spherical harmonics symmetrized by parity) is needed to verify they permit the subset decomposition while respecting ∑x_i²=1.

minor comments (2)

Clarify the notation for the parity vector ξ early in the paper.
[Introduction] Add a reference to classical ANOVA decomposition for comparison in the introduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and have revised the manuscript to incorporate the requested clarifications and proofs.

read point-by-point responses

Referee: [Abstract and §2] The completeness of the decomposition, i.e., that the sum of all f_{u,ξ} recovers f exactly under the spherical measure, is asserted but requires an explicit proof that the chosen orthogonal basis induces projections that are complete in L²(S^d) and that no cross terms arise from the coordinate coupling; without this, the claim that each term depends only on variables in u may not hold uniquely.

Authors: We agree that an explicit proof of completeness strengthens the manuscript. In the revised version, Section 2 now contains a detailed proof that the parity-augmented orthogonal projections are complete in L²(S^d) with respect to the spherical measure. The argument proceeds by verifying that the chosen basis (restrictions of multivariate polynomials orthogonalized on the sphere) spans the full space and that the parity decomposition eliminates cross terms arising from the quadratic constraint ∑x_i²=1, ensuring each f_{u,ξ} depends only on the coordinates in u. revision: yes
Referee: [§4] The specific orthogonal basis functions are referred to but not constructed in detail; an example or definition (such as restrictions of multivariate polynomials or spherical harmonics symmetrized by parity) is needed to verify they permit the subset decomposition while respecting ∑x_i²=1.

Authors: We have expanded Section 4 with an explicit construction of the basis functions as restrictions to the sphere of even- and odd-degree multivariate polynomials, orthogonalized via Gram-Schmidt with respect to the surface measure. We also provide a concrete low-dimensional example (d=2) using parity-symmetrized spherical harmonics to demonstrate how the subset decomposition is realized while automatically satisfying the sphere constraint. revision: yes

Circularity Check

0 steps flagged

No circularity: decomposition grounded in spherical geometry and orthogonal bases

full rationale

The provided abstract and description present formulas for an ANOVA-style decomposition on the sphere that incorporates parity via an additional index ξ, relying on the natural geometry of S^d and unspecified but standard orthogonal basis functions for approximation. No equations or steps are exhibited that reduce the claimed sum of f_{u,ξ} terms to a fitted parameter, self-definition, or self-citation chain; the completeness claim is asserted via the choice of bases without reducing to the input data by construction. This is the expected non-finding for a paper whose central construction is an explicit orthogonal expansion under the spherical measure.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard properties of the sphere and orthogonal expansions rather than new postulates.

axioms (2)

standard math Orthogonal basis functions exist on the sphere that allow the stated decomposition
Invoked for function approximation in the abstract
domain assumption Sphere geometry induces natural dependencies among coordinates
Explicitly stated as motivation for the decomposition

pith-pipeline@v0.9.0 · 5432 in / 1109 out tokens · 39141 ms · 2026-05-16T19:17:21.819647+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean, IndisputableMonolith/Cost/FunctionalEquation.lean alexander_duality_circle_linking, washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We propose a spherical ANOVA decomposition ... using the decomposition of a function into functions with different parity, which adds the additional parameter ξ ... orthogonal basis functions ... Gegenbauer polynomials ... Jacobi polynomials
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective, embed_strictMono_of_one_lt unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The terms A_u f and A_v f ... are orthogonal with respect to the surface measure on S^d

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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Abramowitz and I

M. Abramowitz and I. A. Stegun.Handbook of Mathematical Functions, With Formulas, Graphs, and Mathe- matical Tables. National Bureau of Standards, 1964

work page 1964

[2] [2]

Bungartz and M

H.-J. Bungartz and M. Griebel. Sparse grids.Acta Numer., 13:147–269, 2004.doi:10.1017/ S0962492904000182

work page 2004

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Caflisch, W

R. Caflisch, W. Morokoff, and A. Owen. Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension.J. Comput. Finance, 1(1):27–46, 1997.doi:10.21314/jcf.1997.005

work page doi:10.21314/jcf.1997.005 1997

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M. Chu, N. Del Buono, L. Lopez, and T. Politi. On the low-rank approximation of data on the unit sphere.SIAM Journal on Matrix Analysis and Applications, 27(1):46–60, Jan. 2005.doi:10.1137/s0895479803433295

work page doi:10.1137/s0895479803433295 2005

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da Veiga

S. da Veiga. Kernel-based ANOV A decomposition and shapley effects – application to global sensitivity analysis, 2021.arXiv:2101.05487,doi:10.48550/ARXIV.2101.05487. 24 Sensitivity Analysis on the Sphere and a Spherical ANOV A DecompositionA PREPRINT

work page doi:10.48550/arxiv.2101.05487 2021

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Dai and Y

F. Dai and Y . Xu.Approximation Theory and Harmonic Analysis on Spheres and Balls. 01 2013.doi:10.1007/ 978-1-4614-6660-4

work page 2013

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C. F. Dunkl and Y . Xu.Orthogonal Polynomials of Several Variables. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2nd edition, 2014.doi:10.1017/CBO9781107786134

work page doi:10.1017/cbo9781107786134 2014

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URL https://projecteuclid.org/euclid.aos/1176345802

B. Efron and C. Stein. The Jackknife Estimate of Variance.Ann. Stat., 9(3):586–596, 1981.doi:10.1214/ aos/1176345462

work page arXiv 1981

[9] [9]

Filbir and W

F. Filbir and W. Themistoclakis. Polynomial approximation on the sphere using scattered data.Math. Nachr., 281:650–668, 2008.doi:10.1002/mana.200710633

work page doi:10.1002/mana.200710633 2008

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Griebel and M

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work page doi:10.1016/j.jco.2010.06.001 2010

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M. Hegland. Adaptive sparse grids. InCTAC2001, volume 44, pages C335–C353, 2003.doi:10.21914/ anziamj.v44i0.685

work page 2003

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30 Leland McInnes, John Healy, and Steve Astels

W. Hoeffding. A class of statistics with asymptotically normal distribution.Ann. Math. Statist., 19(3):293–325, 1948.doi:10.1214/aoms/1177730196

work page doi:10.1214/aoms/1177730196 1948

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Holtz.Sparse grid quadrature in high dimensions with applications in finance and insurance, volume 77 ofLecture Notes in Computational Science and Engineering

M. Holtz.Sparse grid quadrature in high dimensions with applications in finance and insurance, volume 77 ofLecture Notes in Computational Science and Engineering. Springer-Verlag, Berlin, 2011.doi:10.1007/ 978-3-642-16004-2

work page 2011

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G. Hooker. Generalized functional ANOV A diagnostics for high-dimensional functions of dependent variables. J. Comput. Graph. Statist., 16(3):709–732, 2007. URL:http://www.jstor.org/stable/27594267

work page arXiv 2007

[16] [16]

Jiang and A

T. Jiang and A. B. Owen. Quasi-regression with shrinkage.Math. Comput. Simul., 62(3):231–241, 2003. 3rd IMACS Seminar on Monte Carlo Methods.doi:10.1016/S0378-4754(02)00253-7

work page doi:10.1016/s0378-4754(02)00253-7 2003

[17] [17]

Kucherenko, S

S. Kucherenko, S. Tarantola, and P. Annoni. Estimation of global sensitivity indices for models with dependent variables.Comput. Phys. Commun., 183(4):937–946, 2012.doi:10.1016/j.cpc.2011.12.020

work page doi:10.1016/j.cpc.2011.12.020 2012

[18] [18]

F. Y . Kuo, I. H. Sloan, G. W. Wasilkowski, and H. Wo´zniakowski. On decompositions of multivariate functions. Math. Comp., 79(270):953–966, 2009.doi:10.1090/s0025-5718-09-02319-9

work page doi:10.1090/s0025-5718-09-02319-9 2009

[19] [19]

Labreuche.Contribution of Subsets of Variables in Global Sensitivity Analysis with Dependent Variables, pages 233–248

C. Labreuche.Contribution of Subsets of Variables in Global Sensitivity Analysis with Dependent Variables, pages 233–248. Springer Nature Switzerland, Nov. 2024.doi:10.1007/978-3-031-76235-2_18

work page doi:10.1007/978-3-031-76235-2_18 2024

[20] [20]

Lemieux and A

C. Lemieux and A. B. Owen. Quasi-regression and the relative importance of the ANOV A components of a function. InMCQMC 2000, pages 331–344. Springer Berlin Heidelberg, 2002.doi:10.1007/ 978-3-642-56046-0_22

work page 2000

[21] [21]

Li and H

G. Li and H. Rabitz. General formulation of HDMR component functions with independent and correlated variables.J. Math. Chem., 50:99–130, 2012.doi:10.1007/s10910-011-9898-0

work page doi:10.1007/s10910-011-9898-0 2012

[22] [22]

Lippert, D

L. Lippert, D. Potts, and T. Ullrich. Fast hyperbolic wavelet regression meets ANOV A.Numer. Math., 154:155– 207, 2023.doi:10.1007/s00211-023-01358-8

work page doi:10.1007/s00211-023-01358-8 2023

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