Generalized Functional ANOVA: A Complete Theoretical Framework

Abstract

The functional ANOVA provides a fundamental representation of square-integrable multivariate functions into main effects and higher-order interactions. For independent inputs, the components belong to mutually orthogonal Hilbert subspaces and admit an explicit representation. For dependent inputs, however, the components are only hierarchically orthogonal: although existence and uniqueness results are available, the Hilbert subspaces underlying the generalized decomposition have remained implicit. We resolve this representation problem for continuous inputs supported on a bounded hyperrectangle whose joint density is bounded above and away from zero. We introduce a distribution-adapted family of functions and prove that it forms a Riesz basis of the $L^2$ space, thereby guaranteeing a unique, stable, and unconditionally convergent representation. We then show that, for every coalition of variables, the corresponding block of this basis exactly characterizes the Hilbert subspace containing the functional ANOVA component. Our construction recovers the classical orthogonal decomposition under input independence. As a direct consequence, computing the generalized functional ANOVA reduces to estimating coefficients in an explicit, distribution-adapted basis. Finally, as a \emph{proof of concept}, we introduce an elementary, fast and model-agnostic estimator based on our theoretical results. Experiments on synthetic and real-world datasets illustrate its connections with established tabular explanation methods and show that low order components often capture most of the signal in the model output.