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arxiv: 2605.18422 · v1 · pith:ESGERUKYnew · submitted 2026-05-18 · 📊 stat.ML · cs.LG· math.ST· stat.TH

Generalized Functional ANOVA in Closed-Form: A Unified View of Additive Explanations

Baptiste Ferrere , Nicolas Bousquet , Fabrice Gamboa , Jean-Michel Loubes This is my paper

Pith reviewed 2026-05-19 23:53 UTC · model grok-4.3

classification 📊 stat.ML cs.LGmath.STstat.TH
keywords generalized functional ANOVARiesz basisadditive explanationsdependent inputsHilbert spacemodel interpretabilitySHAPgeneralized additive models
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The pith

Hilbert space methods yield an explicit Riesz basis for generalized functional ANOVA on continuous dependent inputs

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

The paper establishes a way to decompose any model prediction into additive main effects and higher-order interaction terms for continuous input variables even when those variables are statistically dependent. It achieves this by merging Hilbert space techniques with the generalized functional ANOVA to produce a Riesz basis that makes the entire decomposition explicit and directly computable. A sympathetic reader cares because dependence is the normal case in real data, yet most classical tools for interpretability either assume independence or resort to costly approximations. The construction recovers the familiar orthogonal decomposition as a special case when inputs are independent. From the same representation the authors derive a simple algorithm that estimates the decomposition directly from finite samples without requiring knowledge of the underlying model.

Core claim

By combining Hilbert space methods with the generalized functional ANOVA, we build an explicit decomposition Riesz Basis allowing to easily compute the decomposition. Our formulation recovers the classical independent case and its associated orthogonal decomposition. Building on this representation, we propose a simple but mighty algorithm to estimate the decomposition from a data sample in a model-agnostic setting and we compare it empirically with several state-of-the-art explanation methods, demonstrating the power of the approach.

What carries the argument

the explicit decomposition Riesz Basis constructed in the Hilbert space of the input measure, which represents every term of the generalized functional ANOVA as an inner product against basis elements

If this is right

  • The decomposition can be evaluated in closed form once the basis coefficients are known, without numerical integration over the joint distribution.
  • The independent-input orthogonal ANOVA appears as the special case in which the Riesz basis reduces to the usual product basis.
  • A model-agnostic estimator follows immediately by replacing population inner products with their empirical counterparts on a sample.
  • The same representation connects the decomposition to SHAP values and to the terms of a generalized additive model.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Choosing particular Riesz bases, such as orthogonal polynomials with respect to the marginal measures, could yield fast algorithms that scale to moderate dimensions.
  • The construction supplies a theoretical justification for applying additive explanation methods to correlated features without first transforming the data to independence.
  • If the Riesz basis can be adapted to mixed continuous-discrete inputs, the same closed-form route would extend to a wider class of tabular data problems.

Load-bearing premise

The input variables are continuous and the underlying Hilbert space admits a suitable Riesz basis so that the decomposition stays explicit and can be estimated from finite samples.

What would settle it

Generate synthetic continuous data with known dependence structure, compute the true generalized functional ANOVA terms by direct integration, then apply the proposed estimator and check whether the recovered terms match the true decomposition to within sampling error.

Figures

Figures reproduced from arXiv: 2605.18422 by Baptiste Ferrere, Fabrice Gamboa, Jean-Michel Loubes, Nicolas Bousquet.

Figure 1
Figure 1. Figure 1: Estimated main effects on California Housing: our method (black) vs TreeHFD (main effects) and TreeSHAP on a trained XGB. actually quantify. Second, they are frequently computationally expensive, and in some cases formally intractable. However, these constructions are in fact closely connected to functional decompositions of the predictor. Highlighting this connection has two important consequences. First,… view at source ↗
Figure 2
Figure 2. Figure 2: Decomposition of a trained MLP on Bike Sharing. Left: Network plot for a random instance of the dataset for visualizing local feature attribution and interaction. Middle & Right: For the features hour and atemp our method (black) vs KernelSHAP and DeepSHAP. and higher-order effects, producing richer decompositions of the predictor. In parallel, generalized additive models themselves have long been used to … view at source ↗
Figure 3
Figure 3. Figure 3: Estimated main effects for Age on Census Income. Left: Our method (black) vs KernelSHAP and DeepSHAP on a trained MLP. Right: Our method (black) vs TreeHFD (main effects) and TreeSHAP on a trained XGB. 3 Background The Functional ANOVA decomposition provides a mathematical framework for decomposing a real-valued square integrable function ν(X) into a sum of components of increasing order: ν(X) = ν∅ + Xp i=… view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of native main effects from an EBM and a NAM with those recovered by our [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Force plot of positive and negative contributions (main effects, pair effects and residual) for a random instance of Electrical Grid. [18] to continuous random variables. This mechanism will be the key tool to obtain hierarchical orthogonality. First, let ξ∅ := 1. Definition 4.1. For S ⊆ [p], let denote fS the marginal density of XS and define mS := (mj )j∈S ∈ N |S| + . For x ∈ [−1, 1]p we set ξ (mS ) S (x… view at source ↗
Figure 6
Figure 6. Figure 6: Illustration of our method to estimate the [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Estimated main effects in the analytical setting for [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Estimated main effects in the analytical setting for [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Estimated main effects in the unbounded-density setting for [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison between the theoretical ANOVA components and our estimator ( [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Estimated main effects on Bike Sharing: our method (black) vs TreeHFD (main effects) and TreeSHAP on a trained XGB. 0 5 10 15 20 200 100 0 100 200 300 hr 0 1 2 3 4 5 6 150 100 50 0 50 100 weekday 0.0 0.2 0.4 0.6 0.8 1.0 100 80 60 40 20 0 20 holiday 1.0 1.5 2.0 2.5 3.0 3.5 4.0 60 40 20 0 20 40 60 season 0.0 0.2 0.4 0.6 0.8 1.0 150 100 50 0 50 100 atemp 0.0 0.2 0.4 0.6 0.8 1.0 80 60 40 20 0 20 40 60 hum 0.0… view at source ↗
Figure 12
Figure 12. Figure 12: Estimated main effects on Bike Sharing: our method (black) vs KernelSHAP and DeepSHAP on a trained MLP. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Comparison of native main effects from an EBM with those recovered by our method on [PITH_FULL_IMAGE:figures/full_fig_p029_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Estimated main effects on California Housing: our method (black) vs KernelSHAP and DeepSHAP on a trained MLP. 2 4 6 8 10 0.8 0.0 0.8 1.6 2.4 MedInc 15 30 45 0.30 0.15 0.00 0.15 0.30 HouseAge 4 6 8 10 0.0 2.5 5.0 7.5 10.0 AveRooms 0.9 1.2 1.5 1.8 2.1 7.5 5.0 2.5 0.0 AveBedrms 0 1500 3000 4500 0.08 0.00 0.08 Population 2 3 4 5 0.8 0.0 0.8 1.6 AveOccup 34 36 38 40 2 1 0 1 Latitude 123.0 121.5 120.0 118.5 117… view at source ↗
Figure 15
Figure 15. Figure 15: Comparison of native main effects from an EBM with those recovered by our method on [PITH_FULL_IMAGE:figures/full_fig_p029_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Estimated main effects on Census Income: our method (black) vs TreeHFD (main effects) and TreeSHAP on a trained XGB. 20 30 40 50 60 70 80 90 3 2 1 0 1 2 age 0 2 4 6 8 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 workclass 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1e6 0.4 0.2 0.0 0.2 0.4 0.6 fnlwgt 0 2 4 6 8 10 12 14 1.0 0.5 0.0 0.5 1.0 1.5 2.0 education 2 4 6 8 10 12 14 16 2 1 0 1 2 education-num 0 1 2 3 4 5 6 2.0 1.5 1.0 0.5 0… view at source ↗
Figure 17
Figure 17. Figure 17: Estimated main effects on Census Income: our method (black) vs KernelSHAP and DeepSHAP on a trained MLP. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Comparison of native main effects from an EBM with those recovered by our method on [PITH_FULL_IMAGE:figures/full_fig_p031_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Estimated main effects on Electrical Grid: our method (black) vs TreeHFD (main effects) and TreeSHAP on a trained XGB. 31 [PITH_FULL_IMAGE:figures/full_fig_p031_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Estimated main effects on Electrical Grid: our method (black) vs KernelSHAP and DeepSHAP on a trained MLP. 2 4 6 8 10 4 2 0 2 ¿1 2 4 6 8 10 4 2 0 2 ¿2 2 4 6 8 10 4 2 0 2 ¿3 2 4 6 8 10 4 2 0 2 ¿4 2.4 3.2 4.0 4.8 0.00 0.15 0.30 p1 2.0 1.6 1.2 0.8 0.16 0.08 0.00 0.08 p2 2.0 1.6 1.2 0.8 0.06 0.03 0.00 0.03 0.06 p3 2.0 1.6 1.2 0.8 0.00 0.25 0.50 0.75 p4 0.2 0.4 0.6 0.8 1.0 3.0 1.5 0.0 1.5 3.0 °1 0.2 0.4 0.6 0.… view at source ↗
Figure 21
Figure 21. Figure 21: Comparison of native main effects from an EBM with those recovered by our method on [PITH_FULL_IMAGE:figures/full_fig_p032_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Estimated main effects on Diabetes: our method (black) vs TreeHFD (main effects) and TreeSHAP on a trained XGB. 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 0.5 0.0 0.5 1.0 1.5 preg 0 25 50 75 100 125 150 175 200 4 3 2 1 0 1 2 3 plas 0 20 40 60 80 100 120 0.5 0.0 0.5 1.0 1.5 2.0 pres 0 20 40 60 80 100 0.3 0.2 0.1 0.0 0.1 0.2 skin 0 200 400 600 800 0.8 0.6 0.4 0.2 0.0 0.2 insu 0 10 20 30 40 50 60 70 3 2 1 0 1 2 mas… view at source ↗
Figure 23
Figure 23. Figure 23: Estimated main effects on Diabetes: our method (black) vs KernelSHAP and DeepSHAP on a trained MLP. 33 [PITH_FULL_IMAGE:figures/full_fig_p033_23.png] view at source ↗
read the original abstract

The functional ANOVA, or Hoeffding decomposition, provides a principled framework for interpretability by decomposing a model prediction into main effects and higher-order interactions. For independent inputs, this classical decomposition is explicit. It is closely connected to SHAP values, generalized additive models, and orthogonal polynomial expansions, and therefore constitutes a fundamental tool for additive explainability. In the more general and realistic dependent setting, however, obtaining a tractable representation and estimating the decomposition from data remain challenging. In this work, we address this problem for continuous inputs. By combining Hilbert space methods with the generalized functional ANOVA, we build an explicit decomposition Riesz Basis allowing to easily compute the decomposition. Our formulation recovers the classical independent case and its associated orthogonal decomposition. Building on this representation, we propose a simple but mighty algorithm to estimate the decomposition from a data sample in a model-agnostic setting and we compare it empirically with several state-of-the-art explanation methods, demonstrating the power of the approach.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that Hilbert-space methods applied to the generalized functional ANOVA yield an explicit Riesz basis for decomposing model predictions into additive terms and interactions when inputs are continuous and possibly dependent. The resulting representation is asserted to be directly computable, to recover the classical orthogonal Hoeffding decomposition under independence, and to support a simple model-agnostic estimation procedure from finite samples whose performance is compared empirically with existing explanation methods.

Significance. If the explicit Riesz-basis construction is valid and remains tractable for general joint distributions, the work would supply a principled, closed-form unification of additive explanations that extends beyond the independent-input case while preserving connections to SHAP, GAMs, and orthogonal expansions. The empirical comparisons constitute a concrete strength that would help establish practical utility.

major comments (2)
  1. [Abstract and Riesz-basis construction section] Abstract and the section presenting the Riesz-basis construction: the central claim that an explicit, easily computable Riesz basis exists for arbitrary continuous joint distributions is load-bearing, yet the provided outline supplies neither the explicit form of the basis functions nor a demonstration that coefficient extraction avoids solving a Fredholm integral equation whose kernel depends on the unknown density; without this, the 'closed-form' and 'easily compute' guarantees cannot be verified.
  2. [Recovery of independent case] Section on recovery of the independent case: the manuscript must show, via direct substitution or limit argument, that the generalized basis reduces exactly to the classical orthogonal decomposition when the joint measure factors, rather than merely stating recovery at a high level.
minor comments (2)
  1. [Preliminaries] Clarify the precise definition of the underlying Hilbert space L²(μ) and the inner product used to define the Riesz basis, including any regularity conditions on the joint density.
  2. [Experiments] The empirical section would benefit from reporting standard errors or confidence intervals on the explanation metrics to strengthen the comparison with state-of-the-art methods.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments. These have helped us identify areas where the manuscript would benefit from greater explicitness and rigor. We address each major comment below and have revised the manuscript to incorporate the requested details, proofs, and clarifications.

read point-by-point responses

  1. Referee: [Abstract and Riesz-basis construction section] Abstract and the section presenting the Riesz-basis construction: the central claim that an explicit, easily computable Riesz basis exists for arbitrary continuous joint distributions is load-bearing, yet the provided outline supplies neither the explicit form of the basis functions nor a demonstration that coefficient extraction avoids solving a Fredholm integral equation whose kernel depends on the unknown density; without this, the 'closed-form' and 'easily compute' guarantees cannot be verified.

    Authors: We agree that the original presentation would be strengthened by a more detailed derivation of the Riesz basis. In the revised manuscript we now include an explicit construction: the basis functions are the Riesz representers of the coordinate functionals on the subspaces of the generalized ANOVA decomposition with respect to the joint measure. These are obtained via a Gram-Schmidt orthogonalization that exploits the nested structure of the ANOVA subspaces, yielding closed-form expressions involving only the joint density evaluated at the observed points and the marginal conditionals. We further show that the expansion coefficients are inner products that reduce to expectations under the data-generating distribution; these expectations are estimated directly from samples via Monte Carlo averages and do not require solving any integral equation whose kernel involves the unknown density. A new subsection with the full derivation and a worked example for the bivariate case has been added. revision: yes

  2. Referee: [Recovery of independent case] Section on recovery of the independent case: the manuscript must show, via direct substitution or limit argument, that the generalized basis reduces exactly to the classical orthogonal decomposition when the joint measure factors, rather than merely stating recovery at a high level.

    Authors: We thank the referee for this precise request. The revised manuscript now contains a dedicated lemma with a direct substitution argument. When the joint measure factors as the product of the marginals, the inner-product structure of the generalized Riesz basis collapses to the standard L² inner product with respect to the product measure. Substituting the product form into the defining equations for the basis functions shows that they coincide exactly with the classical orthogonal polynomials (or indicator functions) of the Hoeffding decomposition. The coefficients likewise reduce to the usual centered conditional expectations. The proof is presented in full, including the verification that all cross terms vanish under independence. revision: yes

Circularity Check

0 steps flagged

Derivation is a direct Hilbert-space construction with no reduction to inputs by construction

full rationale

The paper constructs an explicit Riesz basis for the generalized functional ANOVA by combining standard Hilbert-space methods with the existing generalized ANOVA framework for continuous inputs. The abstract states that this yields a tractable decomposition that recovers the classical orthogonal Hoeffding decomposition under independence, which functions as a consistency check rather than an input. No equations or steps are shown that define a quantity in terms of itself, rename a fitted parameter as a prediction, or rely on a load-bearing self-citation whose content is unverified. The central claim remains a mathematical construction from external functional-analysis results and is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the existence of a Riesz basis within a Hilbert space for the generalized functional ANOVA operator when inputs are continuous; this is a standard functional-analysis tool rather than a new postulate, but its applicability to the decomposition is the key modeling choice.

axioms (1)
  • domain assumption Input variables are continuous.
    The work explicitly restricts attention to the continuous-input setting.
invented entities (1)
  • Riesz basis for the generalized functional ANOVA no independent evidence
    purpose: To furnish an explicit, computable representation of the decomposition under input dependence.
    The basis is constructed within the paper as the central technical device; no external empirical handle is supplied in the abstract.

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