Generalized Functional ANOVA

• Generalized Functional ANOVA is a measure-dependent extension of functional ANOVA that uniquely decomposes multivariate functions into lower-dimensional components under sum-to-zero constraints.
• It provides a unified framework for orthogonal or hierarchically orthogonal decompositions, facilitating applications in global sensitivity analysis, interpretable machine learning, and semiparametric efficiency.
• The method employs specific basis constructions and purification techniques to ensure unique, stable, and computationally tractable decompositions even with dependent input distributions.

Generalized Functional ANOVA is a measure-dependent extension of the functional ANOVA, or Hoeffding decomposition, in which a multivariate function is written as a sum of lower-dimensional component functions—main effects, pairwise interactions, and higher-order interactions—subject to identifiability constraints defined relative to a chosen reference measure. In the modern literature, this framework serves both as a theoretical device for orthogonal or hierarchically orthogonal decomposition and as a practical foundation for interpretable machine learning, global sensitivity analysis, semiparametric efficiency, and nonparametric inference (Park et al., 21 Feb 2025).

1. Definition, decomposition, and identifiability

In its standard form, functional ANOVA writes a function $f:\mathcal{X}\to\mathbb{R}$, with $$\mathcal{X}=\mathcal{X}_1\times\cdots\times\mathcal{X}_p$$, as

$$f(x) = \beta_0 + \sum_{S\subseteq[p],\,|S|\ge 1} f_S(x_S),$$

or, in truncated form,

$$f(x) = \beta_0 + \sum_{S\subseteq [p],\, |S|\le d} f_S(x_S).$$

Special cases include GAM for $d=1$ and GA$^2$M for main effects plus pairwise interactions (Park et al., 21 Feb 2025).

The generalized viewpoint emphasizes that such a decomposition is not unique unless additional constraints are imposed. A standard formulation fixes a reference measure $\mu=\prod_j \mu_j$ and requires each component $f_S$ to satisfy the sum-to-zero condition

$$\forall j\in S,\quad \forall z\in \mathcal{X}_{S\setminus\{j\}},\qquad \int_{\mathcal{X}_j} f_S(x_S)\,\mu_j(dx_j)=0.$$

Equivalently, every component has mean zero in each argument with respect to the chosen marginal measure. Under these constraints, the decomposition is unique almost everywhere with respect to $\mu$ (Park et al., 21 Feb 2025).

A closely related formulation uses orthogonality with respect to a weighting distribution $$\mathcal{X}=\mathcal{X}_1\times\cdots\times\mathcal{X}_p$$0. In the weighted $$\mathcal{X}=\mathcal{X}_1\times\cdots\times\mathcal{X}_p$$1 setting, functional ANOVA can be defined as the solution of a least-squares problem over component spaces $$\mathcal{X}=\mathcal{X}_1\times\cdots\times\mathcal{X}_p$$2, subject to orthogonality against all lower-order subspaces. Hooker’s formulation shows that these orthogonality constraints are equivalent to integrate-to-zero conditions, and under nondegeneracy assumptions the resulting decomposition is unique (Lengerich et al., 2019).

For dependent inputs, generalized functional ANOVA replaces the classical product-measure orthogonality by a decomposition defined relative to the actual joint distribution or to a selected measure. This is the sense in which the decomposition is generalized in the dependent-variable setting, in empirical-design settings, and in model-based interpretability frameworks (Park et al., 3 Sep 2025).

2. Reference measures, dependence, and alternative generalized decompositions

A defining feature of generalized functional ANOVA is dependence on the reference measure. Different choices of $$\mathcal{X}=\mathcal{X}_1\times\cdots\times\mathcal{X}_p$$3 lead to different decompositions. In neural and Bayesian implementations, $$\mathcal{X}=\mathcal{X}_1\times\cdots\times\mathcal{X}_p$$4 may be the empirical distribution of the covariates, a uniform distribution on the support, or another user-specified product measure (Park et al., 21 Feb 2025). In purification methods for tree and piecewise-constant models, the weight $$\mathcal{X}=\mathcal{X}_1\times\cdots\times\mathcal{X}_p$$5 may be uniform, empirical, or Laplace-smoothed, and the paper emphasizes that different choices of $$\mathcal{X}=\mathcal{X}_1\times\cdots\times\mathcal{X}_p$$6 yield different purified decompositions (Lengerich et al., 2019).

This dependence is explicit in ANOVA-BART, which uses the empirical distribution $$\mathcal{X}=\mathcal{X}_1\times\cdots\times\mathcal{X}_p$$7 of the design points as the reference measure for identifiability, while allowing the population design density $$\mathcal{X}=\mathcal{X}_1\times\cdots\times\mathcal{X}_p$$8 in the theoretical analysis to be non-product under a Radon–Nikodym bound relative to the product of marginals (Park et al., 3 Sep 2025). It is also explicit in explanation frameworks that distinguish baseline, marginal, and conditional value functions: $$\mathcal{X}=\mathcal{X}_1\times\cdots\times\mathcal{X}_p$$9 These produce baseline, marginal, and conditional fANOVA decompositions, respectively, and coincide only under specific conditions such as linearity or input independence (Fumagalli et al., 2024).

A related generalization appears in uncertainty quantification. For possibly dependent inputs, one may introduce a representation

$$f(x) = \beta_0 + \sum_{S\subseteq[p],\,|S|\ge 1} f_S(x_S),$$0

with $$f(x) = \beta_0 + \sum_{S\subseteq[p],\,|S|\ge 1} f_S(x_S),$$1, and apply functional ANOVA to the induced function $$f(x) = \beta_0 + \sum_{S\subseteq[p],\,|S|\ge 1} f_S(x_S),$$2. This reduction allows first-order and total ANOVA functionals to be defined even when the original inputs are dependent (Lamboni, 2023).

The same measure-dependence appears in model explanation based on modes rather than means. In that setting, the ANOVA components are classical Sobol’/Hoeffding terms defined through conditional expectations under the joint distribution $$f(x) = \beta_0 + \sum_{S\subseteq[p],\,|S|\ge 1} f_S(x_S),$$3, but they are used to compare an observation with a MAP point associated with a target mode of the label distribution (Long, 2024).

3. Representation, bases, and computational constructions

Generalized functional ANOVA has recently acquired explicit constructive forms in both categorical and continuous settings. For categorical inputs, the paper "Exact Functional ANOVA Decomposition for Categorical Inputs Models" derives a closed-form decomposition using a generalized parity basis

$$f(x) = \beta_0 + \sum_{S\subseteq[p],\,|S|\ge 1} f_S(x_S),$$4

and shows that every $$f(x) = \beta_0 + \sum_{S\subseteq[p],\,|S|\ge 1} f_S(x_S),$$5 admits an expansion

$$f(x) = \beta_0 + \sum_{S\subseteq[p],\,|S|\ge 1} f_S(x_S),$$6

from which the component functions $$f(x) = \beta_0 + \sum_{S\subseteq[p],\,|S|\ge 1} f_S(x_S),$$7 are obtained by grouping terms with the same subset $$f(x) = \beta_0 + \sum_{S\subseteq[p],\,|S|\ge 1} f_S(x_S),$$8 (Ferrere et al., 3 Mar 2026). In the Boolean uniform case, this basis reduces, up to scaling, to the classical parity basis of discrete Fourier analysis (Ferrere et al., 3 Mar 2026).

For continuous inputs, the paper "Generalized Functional ANOVA in Closed-Form: A Unified View of Additive Explanations" constructs an explicit decomposition by introducing inverse-likelihood-weighted Legendre basis functions

$$f(x) = \beta_0 + \sum_{S\subseteq[p],\,|S|\ge 1} f_S(x_S),$$9

and proves that the resulting family is a Riesz basis of $$f(x) = \beta_0 + \sum_{S\subseteq [p],\, |S|\le d} f_S(x_S).$$0. The generalized ANOVA components are then represented as

$$f(x) = \beta_0 + \sum_{S\subseteq [p],\, |S|\le d} f_S(x_S).$$1

which yields a closed-form characterization of Hooker’s weighted ANOVA decomposition in the continuous dependent-input setting (Ferrere et al., 18 May 2026).

A different constructive approach appears in tree-based purification. Piecewise-constant models are expressed as sums of tensors $$f(x) = \beta_0 + \sum_{S\subseteq [p],\, |S|\le d} f_S(x_S).$$2, and a mass-moving algorithm iteratively subtracts weighted slice means from higher-order tensors and adds them to lower-order tensors. This produces a purified canonical representation satisfying the functional ANOVA constraints exactly while preserving the original predictions (Lengerich et al., 2019).

For smoothing spline ANOVA, the decomposition is built through tensor-product Sobolev spaces and basis transformations. In doubly penalized ANOVA modeling, hierarchical total variation is constructed so that, after changing basis, the penalty becomes an $$f(x) = \beta_0 + \sum_{S\subseteq [p],\, |S|\le d} f_S(x_S).$$3 penalty on spline coefficients, and the ANOVA decomposition becomes a block decomposition of the coefficient vector (Yang et al., 2019). In effect-wise inference for smoothing spline ANOVA, each effect space $$f(x) = \beta_0 + \sum_{S\subseteq [p],\, |S|\le d} f_S(x_S).$$4 is an RKHS, and the full kernel decomposes as

$$f(x) = \beta_0 + \sum_{S\subseteq [p],\, |S|\le d} f_S(x_S).$$5

which makes the estimator and its asymptotics effect-specific (Cho et al., 2 Feb 2026).

Neural constructions encode identifiability directly in the architecture. ANOVA-TPNN uses centered one-dimensional basis networks

$$f(x) = \beta_0 + \sum_{S\subseteq [p],\, |S|\le d} f_S(x_S).$$6

with

$$f(x) = \beta_0 + \sum_{S\subseteq [p],\, |S|\le d} f_S(x_S).$$7

and builds interaction bases as tensor products

$$f(x) = \beta_0 + \sum_{S\subseteq [p],\, |S|\le d} f_S(x_S).$$8

This guarantees that every component satisfies the generalized ANOVA constraints by construction (Park et al., 21 Feb 2025). Bayesian-TPNN extends this idea by making the subsets $$f(x) = \beta_0 + \sum_{S\subseteq [p],\, |S|\le d} f_S(x_S).$$9 random and learning higher-order interactions through a Bayesian structure-learning procedure (Park et al., 1 Oct 2025).

4. Statistical theory, approximation, and inference

Generalized functional ANOVA supports several layers of statistical theory: approximation, posterior contraction, efficient estimation, and effect-wise frequentist inference.

For ANOVA-TPNN, approximation results are stated for Lipschitz component functions satisfying the generalized ANOVA constraints. If $d=1$0 is $d=1$1-Lipschitz, then there exists a tensor-product neural representation with uniform error

$d=1$2

For a full GA$d=1$3M, the total approximation error is bounded by a sum of these terms over all $d=1$4 with $d=1$5 (Park et al., 21 Feb 2025). The dependence on $d=1$6 makes explicit that higher-order interactions require more basis functions (Park et al., 21 Feb 2025).

In ANOVA-BART, each component $d=1$7 is represented as a sum of identifiable binary-product trees, and the posterior concentrates at a near-minimax rate for the full function and, more unusually, for each component separately. If

$d=1$8

then the posterior contracts around $d=1$9 at rate $^2$0, and for every $^2$1,

$^2$2

in $^2$3-probability (Park et al., 3 Sep 2025).

Bayesian-TPNN establishes posterior consistency for both the full regression function and each individual ANOVA component under a unique decomposition with sum-to-zero constraints (Park et al., 1 Oct 2025). This places effect recovery, not only prediction, within a Bayesian asymptotic framework.

In smoothing spline ANOVA on tensor-product Sobolev spaces, effect-wise inference is formulated directly at the level of the component spaces $^2$4. The orthogonality decomposition of effect subspaces allows a functional Bahadur representation to be extended to each $^2$5, yielding convergence rates, pointwise confidence intervals, and a Wald-type test for $^2$6 (Cho et al., 2 Feb 2026). Main effects achieve optimal univariate rates, while interactions achieve optimal rates up to logarithmic factors (Cho et al., 2 Feb 2026).

Generalized functional ANOVA also appears in semiparametric efficiency. In incomplete-data problems with multiple observation patterns $^2$7, the asymptotic minimal mean squared error for estimating a linear functional $^2$8 is given by the minimum of a quadratic functional

$^2$9

The minimizer defines a generalized ANOVA decomposition of the efficient influence function, and the resulting estimator attains the corresponding semiparametric efficiency bound asymptotically (Berrett, 2024).

5. Interpretability, sensitivity analysis, and cooperative game theory

Generalized functional ANOVA has become a unifying language for interpretability. In local explanation games, the Möbius transform of a set function built from ANOVA value functions equals the fANOVA effect itself: $\mu=\prod_j \mu_j$0 This identifies each ANOVA component as a Harsanyi dividend of a cooperative game, and Shapley values or interaction indices arise by reweighting these component functions (Fumagalli et al., 2024).

For a GA$\mu=\prod_j \mu_j$1M satisfying the sum-to-zero condition, the Shapley value of feature $\mu=\prod_j \mu_j$2 admits the representation

$\mu=\prod_j \mu_j$3

This formula, highlighted in neural ANOVA work, implies that the contribution of interaction $\mu=\prod_j \mu_j$4 to feature $\mu=\prod_j \mu_j$5’s SHAP value is $\mu=\prod_j \mu_j$6 (Park et al., 21 Feb 2025). In the categorical closed-form setting, the same formula is proposed as a natural generalization of SHAP to arbitrary dependent categorical inputs (Ferrere et al., 3 Mar 2026).

Global sensitivity analysis provides another interpretation. For independent inputs, the variance decomposition

$\mu=\prod_j \mu_j$7

recovers Sobol’ indices as normalized variance contributions (Fumagalli et al., 2024). In the sensitivity-game formulation, the Möbius transform is $\mu=\prod_j \mu_j$8 in the independent case, while under dependence it includes covariance terms between generalized ANOVA components (Fumagalli et al., 2024).

Kernel-based measures of association generalize variance-based ANOVA functionals by embedding first-order and total ANOVA functionals into an RKHS. For a kernel $\mu=\prod_j \mu_j$9, the paper defines normalized kernel-based sensitivity indices

$f_S$0

and shows that, for Gaussian-distributed AFs, quadratic kernels recover Sobol’ indices and dependent generalized sensitivity indices (Lamboni, 2023).

Interpretability also motivates purification and stability. In piecewise-constant models, unpurified main and interaction effects can be contradictory, because the same function may admit multiple parameterizations. Purifying interactions by functional ANOVA yields a canonical representation in which higher-order effects capture only variance not explainable by lower-order subsets (Lengerich et al., 2019). In neural ANOVA models, encoding the generalized ANOVA constraints architecturally leads to much more stable estimation of each component than existing neural networks when training data and initial values vary (Park et al., 21 Feb 2025).

6. Extensions, applications, and limitations

Generalized functional ANOVA now spans a wide range of application domains. In tree-based and tabular prediction, it supports purified interaction effects, tensor-product neural decompositions, Bayesian tree ensembles, and generalized additive explainers (Lengerich et al., 2019). In explanation theory, it unifies perturbation-based, gradient-based, risk-based, and game-theoretic feature attributions by varying the decomposition measure and the aggregation rule over ANOVA components (Fumagalli et al., 2024). In uncertainty quantification, it underlies dependent-input sensitivity analysis and kernel-based dependence measures (Lamboni, 2023). In semiparametric inference, it characterizes efficient influence functions in incomplete-data problems (Berrett, 2024).

The framework also extends beyond standard multivariate regression. In multivariate functional data, a functional GLHT

$f_S$1

encompasses one-way FMANOVA, post hoc tests, and contrast analysis as special cases, thereby turning generalized functional ANOVA into a global inferential tool for vector-valued mean functions under heteroscedastic covariance structures (Zhu, 2023). In warped functional data, amplitude and phase variability are jointly modeled by combining a random-effects functional ANOVA for latent registered curves with random warping functions, yielding amplitude and warping variance ratios $f_S$2 and $f_S$3 (Gervini et al., 2013).

Several limitations recur across the literature. Dependence on the reference measure is intrinsic: changing $f_S$4 changes the decomposition (Park et al., 21 Feb 2025). Product-measure constructions may not explicitly encode feature dependence even when the target application is dependent-input modeling (Park et al., 21 Feb 2025). Higher-order interactions remain computationally challenging, whether in tensor-product approximation, Bayesian subset search, or penalized ANOVA modeling (Park et al., 21 Feb 2025). For continuous closed-form decompositions, bounded-support and bounded-density assumptions play a central role in the Riesz-basis construction (Ferrere et al., 18 May 2026). For categorical closed-form decompositions, non-full support yields overcomplete bases and nonuniqueness unless a basis subfamily is selected (Ferrere et al., 3 Mar 2026).

A plausible implication is that generalized functional ANOVA is best understood not as a single decomposition formula, but as a family of measure-dependent, structure-aware decompositions that specialize to different inferential, algorithmic, and interpretive goals. Across recent work, the common thread is consistent: the decomposition is defined by the interaction between a function space, a reference distribution, and a notion of orthogonality or centering, and modern constructions aim to make that interaction explicit, computable, and inferentially valid (Ferrere et al., 18 May 2026).