Generalized Additive Decomposition (GAD)

• Generalized Additive Decomposition (GAD) is a framework that decomposes complex multivariate functions or tensors into sums of structured, lower-dimensional components.
• It underpins various methodologies including Shapley-GAM in explainable machine learning, functional ANOVA in statistics, and tensor decompositions in algebraic geometry.
• GAD enables clearer interpretation of main effects and interactions in models while offering computational advantages in decision theory and utility modeling.

Generalized Additive Decomposition (GAD) denotes a family of structural representations in which a multivariate object is written as a sum of lower-dimensional components or structured factors. In explainable machine learning, the most natural interpretation is a generalized additive model–type decomposition $f(x)=\sum_{S\subseteq[d]} f_S(x_S)$ that resolves main effects and interactions through Shapley-based value functions or functional ANOVA constructions. In decision theory, closely related decompositions arise in generalized additive independence models. In algebraic geometry, the same expression denotes decompositions of degree-$d$ forms into sums of osculating-type terms $\sum_i \omega_i \ell_i^{\,d-k_i}$. The term is therefore not fully standardized; its common thread is a structured breakup of a complex object into interpretable parts indexed by variables, subsets, or supports (Bordt et al., 2022, Ferrere et al., 18 May 2026, Grabisch et al., 2016, Barrilli et al., 29 Oct 2025).

1. Terminological scope and principal meanings

Across current literature, “Generalized Additive Decomposition” names several mathematically distinct constructions. In some settings it is an explicit term of art; in others it is the most natural description of a construction developed under a different name, such as the “Shapley-GAM” in explainable machine learning (Bordt et al., 2022).

Domain	Canonical form	Representative source
Explainable ML / GAM-type decomposition	$f(x)=\sum_{S\subseteq[d]} f_S(x_S)$	(Bordt et al., 2022, Ferrere et al., 18 May 2026, Idrissi et al., 2023)
Regional decomposition of global effects	$$g(x)\approx \sum_{g=1}^G\sum_{j\in S} f_{j\mid \mathcal A_g}(x_j)\mathbf 1\{x\in\mathcal A_g\}$$	(Herbinger et al., 2023)
Discrete utility / GAI	$U(x)=\sum_{0<|A|\le p} u_A(x_A)$	(Grabisch et al., 2016)
Symmetric tensors	$f=\sum_{i=1}^s \omega_i\,\ell_i^{\,d-k_i}$	(Barrilli et al., 29 Oct 2025)

A recurring source of confusion is acronym collision. “GAD” also abbreviates “Generalized Ambiguity Decomposition,” a loss-dependent ensemble-diversity decomposition, and “Generalized Aliasing Decomposition,” an operator-level decomposition of predictive error. These are not generalized additive decompositions, even though the acronym is the same (Audhkhasi et al., 2013, Transtrum et al., 2024).

2. Shapley-GAM and subset-indexed decompositions of prediction functions

For a prediction function $f:\mathbb R^d\to\mathbb R$, a central GAD form is

$f(x)=\sum_{S\subset[d]} f_S(x_S),$

where $S=\emptyset$ gives a constant term, $d$0 gives main effects, and larger $d$1 give higher-order interactions. In (Bordt et al., 2022), this is tied to a subset-compliant value function $d$2 satisfying $d$3 and $d$4. Observational SHAP and interventional SHAP are the two central examples. The induced component functions are obtained by a Möbius transform,

$d$5

with $d$6. This decomposition is called the Shapley-GAM, and it gives a uniquely determined decomposition once $d$7 is fixed (Bordt et al., 2022).

This construction simultaneously clarifies identifiability and interpretation. Classical GAM-like representations are under-determined unless additional constraints are imposed. The Shapley-GAM resolves that ambiguity by tying the decomposition to the value function through Möbius/Zeta inversion. Conversely, any decomposition $d$8 induces a subset-compliant value function through

$d$9

so the correspondence is exact (Bordt et al., 2022).

The same framework yields a hierarchy of explanation operators. A GAM of order $\sum_i \omega_i \ell_i^{\,d-k_i}$0 is defined by

$\sum_i \omega_i \ell_i^{\,d-k_i}$1

The paper introduces $\sum_i \omega_i \ell_i^{\,d-k_i}$2-Shapley Values, a parametric family of local post-hoc explanations with interaction terms up to order $\sum_i \omega_i \ell_i^{\,d-k_i}$3. At $\sum_i \omega_i \ell_i^{\,d-k_i}$4, they reduce to classical Shapley values; at $\sum_i \omega_i \ell_i^{\,d-k_i}$5, they recover pairwise interaction explanations; at $\sum_i \omega_i \ell_i^{\,d-k_i}$6, they coincide with the Shapley-GAM components $\sum_i \omega_i \ell_i^{\,d-k_i}$7. Under observational SHAP with independent features, or under interventional SHAP without an independence requirement, $\sum_i \omega_i \ell_i^{\,d-k_i}$8-Shapley Values, Shapley–Taylor, and Faith-Shap interaction indices recover the exact GAM components whenever $\sum_i \omega_i \ell_i^{\,d-k_i}$9 is itself a GAM of order $f(x)=\sum_{S\subseteq[d]} f_S(x_S)$0 (Bordt et al., 2022).

A major implication is interpretive rather than merely algebraic. Standard Shapley values are not “interaction-free”; rather, they redistribute higher-order terms into one-dimensional attributions. This gives a principled reading of partial dependence plots of Shapley values: the smooth trend corresponds to the component function $f(x)=\sum_{S\subseteq[d]} f_S(x_S)$1, while residual scatter reflects unresolved interactions. As $f(x)=\sum_{S\subseteq[d]} f_S(x_S)$2 increases, less higher-order structure is pushed into lower-order components, and at $f(x)=\sum_{S\subseteq[d]} f_S(x_S)$3 the decomposition is exact (Bordt et al., 2022).

3. Functional ANOVA, dependent inputs, and uniqueness under hierarchy constraints

A second major lineage treats GAD as a functional ANOVA or Hoeffding-type decomposition. In the independent-input case, a square-integrable function $f(x)=\sum_{S\subseteq[d]} f_S(x_S)$4 admits the classical representation

$f(x)=\sum_{S\subseteq[d]} f_S(x_S)$5

with orthogonality, uniqueness, and explicit Möbius formulas in terms of conditional expectations. The variance then decomposes additively as $f(x)=\sum_{S\subseteq[d]} f_S(x_S)$6 (Ferrere et al., 18 May 2026).

The dependent-input case is more delicate because conditional expectations entangle lower- and higher-order effects. In (Ferrere et al., 18 May 2026), generalized functional ANOVA is defined as the solution of an $f(x)=\sum_{S\subseteq[d]} f_S(x_S)$7 approximation problem over subset-indexed component spaces $f(x)=\sum_{S\subseteq[d]} f_S(x_S)$8, subject to hierarchical orthogonality: $f(x)=\sum_{S\subseteq[d]} f_S(x_S)$9 This makes each $$g(x)\approx \sum_{g=1}^G\sum_{j\in S} f_{j\mid \mathcal A_g}(x_j)\mathbf 1\{x\in\mathcal A_g\}$$0 purely of order $$g(x)\approx \sum_{g=1}^G\sum_{j\in S} f_{j\mid \mathcal A_g}(x_j)\mathbf 1\{x\in\mathcal A_g\}$$1, orthogonal to all functions of strict subsets. Under boundedness assumptions on the density, existence and uniqueness follow. The same paper constructs an explicit decomposition Riesz basis $$g(x)\approx \sum_{g=1}^G\sum_{j\in S} f_{j\mid \mathcal A_g}(x_j)\mathbf 1\{x\in\mathcal A_g\}$$2, built from normalized Legendre polynomials reweighted by inverse marginals $$g(x)\approx \sum_{g=1}^G\sum_{j\in S} f_{j\mid \mathcal A_g}(x_j)\mathbf 1\{x\in\mathcal A_g\}$$3, and shows that every $$g(x)\approx \sum_{g=1}^G\sum_{j\in S} f_{j\mid \mathcal A_g}(x_j)\mathbf 1\{x\in\mathcal A_g\}$$4 has a unique expansion in that basis. It also gives a model-agnostic estimation algorithm based on truncation $$g(x)\approx \sum_{g=1}^G\sum_{j\in S} f_{j\mid \mathcal A_g}(x_j)\mathbf 1\{x\in\mathcal A_g\}$$5, empirical least squares, LassoLarsIC with BIC for sparse basis selection, and SVD for the reduced linear solve. Empirically, the resulting decomposition typically reconstructs at least $$g(x)\approx \sum_{g=1}^G\sum_{j\in S} f_{j\mid \mathcal A_g}(x_j)\mathbf 1\{x\in\mathcal A_g\}$$6–$$g(x)\approx \sum_{g=1}^G\sum_{j\in S} f_{j\mid \mathcal A_g}(x_j)\mathbf 1\{x\in\mathcal A_g\}$$7 of the variance with $$g(x)\approx \sum_{g=1}^G\sum_{j\in S} f_{j\mid \mathcal A_g}(x_j)\mathbf 1\{x\in\mathcal A_g\}$$8, with $$g(x)\approx \sum_{g=1}^G\sum_{j\in S} f_{j\mid \mathcal A_g}(x_j)\mathbf 1\{x\in\mathcal A_g\}$$9 generally around $U(x)=\sum_{0<|A|\le p} u_A(x_A)$0 or lower, and computational time in seconds for tens of thousands of instances (Ferrere et al., 18 May 2026).

A related but distinct formulation in (Idrissi et al., 2023) generalizes Hoeffding decomposition under dependence through probability-theoretic and functional-analytic conditions. The construction defines subspaces $U(x)=\sum_{0<|A|\le p} u_A(x_A)$1 recursively and obtains a direct-sum decomposition

$U(x)=\sum_{0<|A|\le p} u_A(x_A)$2

with canonical components $U(x)=\sum_{0<|A|\le p} u_A(x_A)$3, where $U(x)=\sum_{0<|A|\le p} u_A(x_A)$4 is an oblique projector. The classical conditional-expectation Möbius formula survives only under independence; under dependence it is replaced by an oblique-projection formula

$U(x)=\sum_{0<|A|\le p} u_A(x_A)$5

This yields a canonical additive decomposition with hierarchical orthogonality but not, in general, mutual orthogonality between incomparable subsets. The framework also separates structural, correlative, pure interaction, and dependence effects through variance-based indices derived from the decomposition (Idrissi et al., 2023).

Taken together, these ANOVA-based constructions give the most formal answer to the question of when a GAD is unique: uniqueness is obtained either by subset-compliant Möbius inversion tied to a value function, or by constrained $U(x)=\sum_{0<|A|\le p} u_A(x_A)$6 projection with hierarchical orthogonality. The independent-input formulas are a special case of both (Bordt et al., 2022, Ferrere et al., 18 May 2026, Idrissi et al., 2023).

4. Practical constructions: additive Gaussian processes and regional decompositions of global effects

One practical route to GAD is to encode additivity directly into a probabilistic surrogate model. In (Durrande et al., 2011), an additive kernel for Gaussian process modeling is defined by

$U(x)=\sum_{0<|A|\le p} u_A(x_A)$7

which induces additive sample paths up to modification and yields an additive Kriging mean

$U(x)=\sum_{0<|A|\le p} u_A(x_A)$8

The paper proves that a square-integrable random process with an additive kernel is additive up to a modification, studies univariate submodels and centered effects, and introduces Relaxed Likelihood Maximization (RLM) as a cyclic parameter-estimation procedure with a residual variance term $U(x)=\sum_{0<|A|\le p} u_A(x_A)$9. On Sobol’s $f=\sum_{i=1}^s \omega_i\,\ell_i^{\,d-k_i}$0-function, RLM with an additive Matérn $f=\sum_{i=1}^s \omega_i\,\ell_i^{\,d-k_i}$1 kernel reaches mean $f=\sum_{i=1}^s \omega_i\,\ell_i^{\,d-k_i}$2 with standard deviation $f=\sum_{i=1}^s \omega_i\,\ell_i^{\,d-k_i}$3, compared with $f=\sum_{i=1}^s \omega_i\,\ell_i^{\,d-k_i}$4 for usual joint maximum likelihood with the same additive kernel and $f=\sum_{i=1}^s \omega_i\,\ell_i^{\,d-k_i}$5 for standard non-additive Kriging (Durrande et al., 2011).

A different practical interpretation appears in the decomposition of global feature effects. The GADGET framework—Generalized Additive Decomposition of Global EffecTs—uses recursive partitioning to find regions $f=\sum_{i=1}^s \omega_i\,\ell_i^{\,d-k_i}$6 in which interaction-related heterogeneity of local effects is minimized, so that within each region the joint effect of a feature set $f=\sum_{i=1}^s \omega_i\,\ell_i^{\,d-k_i}$7 is approximated by an additive regional form,

$f=\sum_{i=1}^s \omega_i\,\ell_i^{\,d-k_i}$8

The method is estimator-agnostic for partial dependence, accumulated local effects, and SHAP dependence, provided a local decomposability axiom holds. Its split criterion minimizes within-region variance of local feature effects over candidate split features and thresholds, and a permutation-based interaction test, PINT, is introduced to identify features with significant interaction-related heterogeneity. Under the stated assumptions, the framework minimizes interactions between features of interest and split features, and when no interactions remain, the regional joint effect becomes exactly additive in the regional one-dimensional curves (Herbinger et al., 2023).

These two constructions illustrate complementary uses of GAD. Additive-kernel Gaussian processes impose the decomposition at the model level. GADGET leaves the predictor unchanged and instead decomposes the explanatory view into region-wise additive summaries (Durrande et al., 2011, Herbinger et al., 2023).

5. Discrete utility theory and generalized additive independence

In multicriteria decision theory, a closely related notion appears in generalized additive independence (GAI) models. A utility function on a product space $f=\sum_{i=1}^s \omega_i\,\ell_i^{\,d-k_i}$9 is represented as

$f:\mathbb R^d\to\mathbb R$0

where $f:\mathbb R^d\to\mathbb R$1 may contain overlapping subsets. This generalizes the additive utility model and need not satisfy mutual preferential independence. The difficulty is that GAI decompositions are highly non-unique, and individual terms in a canonical decomposition need not inherit monotonicity properties of the global utility (Grabisch et al., 2016).

The paper (Grabisch et al., 2016) focuses on discrete $f:\mathbb R^d\to\mathbb R$2-additive GAI models. A utility is $f:\mathbb R^d\to\mathbb R$3-additive if variations generated by changing any subset of at most $f:\mathbb R^d\to\mathbb R$4 attributes do not depend on the levels of the remaining attributes, and this is equivalent to a decomposition

$f:\mathbb R^d\to\mathbb R$5

For the $f:\mathbb R^d\to\mathbb R$6-additive case,

$f:\mathbb R^d\to\mathbb R$7

The main result states that every $f:\mathbb R^d\to\mathbb R$8-additive discrete GAI model satisfying monotonicity and boundary assumptions admits an equivalent decomposition into nonnegative monotone terms $f:\mathbb R^d\to\mathbb R$9 and $f(x)=\sum_{S\subset[d]} f_S(x_S),$0. This converts an otherwise ambiguous additive representation into one whose local terms are themselves interpretable as monotone contributions (Grabisch et al., 2016).

The computational significance is substantial. Without the monotone decomposition, monotonicity constraints on $f(x)=\sum_{S\subset[d]} f_S(x_S),$1 scale exponentially with the number of attributes. With the decomposition into monotone singleton and pairwise terms, they scale quadratically. In the homogeneous case with $f(x)=\sum_{S\subset[d]} f_S(x_S),$2 and $f(x)=\sum_{S\subset[d]} f_S(x_S),$3, the number of monotonicity constraints drops from $f(x)=\sum_{S\subset[d]} f_S(x_S),$4 to $f(x)=\sum_{S\subset[d]} f_S(x_S),$5 (Grabisch et al., 2016). In this literature, GAD is therefore not only an interpretive device but also a tractability device.

6. Symmetric tensor GAD and the limits of additive interpretability

In algebraic geometry and tensor decomposition, GAD has a different but structurally analogous meaning. For a degree-$f(x)=\sum_{S\subset[d]} f_S(x_S),$6 form $f(x)=\sum_{S\subset[d]} f_S(x_S),$7, a generalized additive decomposition is

$f(x)=\sum_{S\subset[d]} f_S(x_S),$8

where the $f(x)=\sum_{S\subset[d]} f_S(x_S),$9 are pairwise non-proportional linear forms, the $S=\emptyset$0 are weights, and $S=\emptyset$1 does not divide $S=\emptyset$2. Waring decompositions are the special case $S=\emptyset$3; tangential decompositions correspond to $S=\emptyset$4. Geometrically, each term lies on an osculating variety to the Veronese variety, and a GAD expresses $S=\emptyset$5 as a point on a secant of such varieties (Barrilli et al., 29 Oct 2025).

This tensor notion carries its own rank theory. The paper defines the size of a GAD through local inverse systems, introduces GAD-rank as the minimal achievable size, and proves that under a Castelnuovo–Mumford regularity condition the GAD-rank coincides with the rank of a suitable Catalecticant matrix. Under the same condition, the minimal GAD and the associated apolar scheme are both minimal and unique. A numerical algorithm is then developed from truncated Hankel operators, multiplication matrices, Schur factorization, cluster-based multiplicity recovery, nil-index estimation, and a final linear solve for the weights $S=\emptyset$6 (Barrilli et al., 29 Oct 2025).

At the same time, recent work has questioned whether additive decomposition should remain the default interpretability template in machine learning. “Beyond Additive Decompositions: Interpretability Through Separability” argues that additive representations such as GAMs, SHAP, and functional ANOVA can suffer from signal cancellation and off-support extrapolation in the presence of strong interactions. Its proposed Tensor Separation Learning model instead learns a sum of rank-1 products of univariate functions by a stagewise greedy procedure with orthogonal refitting, and the learned model can be fully reconstructed from first-order partial dependence functions up to constant factors (Liu et al., 29 May 2026). This suggests a natural boundary of GAD: additive decompositions remain central, but they are not the only structured route to interpretability when interaction structure is dominant.

In that broader perspective, GAD occupies a foundational but non-exclusive role. It provides canonical subset-indexed decompositions, dependence-aware ANOVA constructions, regional additive summaries, tractable utility decompositions, and algebraic tensor factorizations. Its unifying idea is decomposition by structure; its main fault line is whether structure should be imposed additively or by some richer geometry of interactions (Bordt et al., 2022, Ferrere et al., 18 May 2026, Barrilli et al., 29 Oct 2025, Liu et al., 29 May 2026).