operator shapley attribution

Traditional Shapley value computation is computationally prohibitive, and existing learnable explanation methods struggle with the non-uniform grids and irregular geometries commonly encountered in physical simulations. This work proposes OperatorSHAP—the first mesh-agnostic attribution method that extends Shapley values to function spaces. By integrating neural operator architectures with a learnable explainer, OperatorSHAP delivers consistent explanations across varying mesh resolutions without requiring model retraining. The method establishes a theoretical connection to the Aumann–Shapley value and demonstrates strong empirical alignment with discrete Shapley values across multiple grid resolutions. Consequently, it significantly enhances both the efficiency and generalization of model interpretability in physics-informed applications.

Shapley values, though theoretically foundational for feature importance interpretation, suffer from exponential computational complexity in the number of features and are further constrained in probabilistic models—such as Gaussian processes—due to the need to model higher-order moments of output distributions. This work addresses these limitations for FANOVA-structured Gaussian processes. We propose the first algorithm that computes exact local and global Shapley value means and variances in *O(d²)* time. Our method introduces a stochastic cooperative game framework with a variance-aware value function that jointly quantifies expected marginal contributions and uncertainty propagation. Leveraging closed-form Möbius inversion, FANOVA decomposition, and a Newton-identity-inspired recursive strategy, we derive efficient analytical solutions. Experiments demonstrate substantial improvements in scalability and interpretability reliability, achieving high accuracy, low computational overhead, and intrinsic uncertainty awareness within structured probabilistic models.

This work addresses the limitations of Shapley values in identifying positive contributors under nonlinear evaluation metrics such as AUC, which stem from their inherent linearity assumption. To overcome this, the paper proposes a nonlinear attribution method that satisfies core axioms—including consistency, equal treatment, and efficiency—by leveraging an optimization framework inspired by the least core to approximate the utility function and yield a unique, optimal contribution vector. This approach transcends the conventional linear attribution paradigm, substantially enhancing attribution reliability while preserving essential axiomatic properties. Experimental results demonstrate that the proposed method outperforms Shapley variants that relax only the efficiency axiom, particularly on AUC-based evaluations, thereby validating the effectiveness and superiority of nonlinear attribution in cooperative settings.

Kernel methods suffer from poor interpretability, and exact Shapley value computation is typically intractable due to exponential time complexity. Method: This paper proposes PKeX-Shapley, the first algorithm enabling exact polynomial-time Shapley value computation under product kernel models. Its core innovation lies in exploiting the multiplicative structure of product kernels to derive a decomposable functional representation and a recursive Shapley value formula, integrating RKHS theory, functional space decomposition, and dynamic programming for efficiency. Contribution/Results: PKeX-Shapley reduces Shapley value computation complexity from exponential to polynomial time, achieving zero-approximation-error attribution in kernel regression and classification. Moreover, it generalizes to statistical discrepancy measures—including MMD and HSIC—enabling rigorous feature-level interpretability analysis while preserving theoretical fidelity to the underlying kernel model.

Traditional Shapley value methods struggle to simultaneously account for externalities among features and exogenous influences, leading to implausible explanations in complex causal structures. This work proposes DAG-SHAP, which introduces edge interventions into the Shapley attribution framework for the first time, treating edges—rather than nodes—as the fundamental units of attribution within a directed acyclic graph (DAG). This finer-grained approach enables more precise characterization of each feature’s role along causal pathways. To ensure scalability, we develop an efficient approximation algorithm and demonstrate through experiments on multiple real-world and synthetic datasets that DAG-SHAP achieves substantially improved attribution accuracy and interpretability compared to existing methods.

This work addresses the longstanding trade-off between accuracy and efficiency in feature- and node-level attribution for graph neural networks (GNNs), which typically rely on numerical approximations of path integrals. The authors propose APEX, a novel framework that introduces PolyGIN—a GNN architecture with a carefully designed polynomial form—enabling, for the first time, exact analytical computation of Aumann-Shapley attribution via path integrals. By integrating Gauss–Legendre quadrature with polynomial message passing, APEX guarantees that model outputs are bounded multivariate polynomials, thereby satisfying both completeness and computational efficiency in attribution. Empirical evaluations demonstrate that APEX maintains strong predictive performance across multiple graph benchmarks while achieving significantly higher attribution fidelity than existing baselines and drastically reducing the number of evaluation points required for path integration.

This work addresses the challenge of efficiently estimating Shapley values in settings where coalition evaluations are computationally expensive and severely budget-constrained. It introduces, for the first time, Bayesian experimental design to this problem by proposing an adaptive sampling method that leverages a Gaussian process surrogate model. The approach strategically selects the most informative coalitions for evaluation by maximizing expected information gain. By exploiting the linearity of Shapley values and properties of elementary symmetric polynomials, the method reduces the computational complexity from exponential to polynomial in the number of players. Empirical results demonstrate that under tight evaluation budgets, the proposed algorithm substantially outperforms existing baselines across multiple high-cost application scenarios, achieving markedly higher sample efficiency.

This work addresses the out-of-manifold artifacts in Shapley value-based explanations caused by heuristic baselines in interpretable AI. The authors propose an axiomatic Aumann-Shapley attribution framework grounded in optimal generative flows, where the baseline path is defined as the Wasserstein-2 geodesic between the baseline and the input. By formulating baseline selection as a variational problem, they derive a unique gradient-based path integral representation that satisfies both efficiency and reparameterization invariance. Theoretical analysis establishes stability bounds on the advection approximation error, ensuring zero flow consistency error and strict manifold consistency. Experiments demonstrate that the proposed method significantly outperforms existing approaches in terms of semantic alignment and structure-aware total variation metrics.

Existing Shapley value–based attribution methods are impractical in composite AI systems due to their requirement to evaluate all subsets of components—a prohibitive task when systems involve unobservable third-party APIs or centralized routing. To address this, this work proposes BOHM, a novel approach that leverages inherent system routing weights to construct a hierarchical attribution tree, enabling multi-granular contribution analysis through the product of root-to-leaf path weights. BOHM achieves zero marginal computational cost and requires no internal system access, offering, for the first time, zero-cost, multi-resolution attribution without re-evaluating the system. Evaluated across 18 language models, 5 drivers, and U.S. Census data, BOHM attains a Kendall’s τ of 0.928—comparable to SHAP’s 0.980 but at 9,000× lower computational overhead—and recovers ground-truth rankings with τ up to 0.722 across resolution levels.

This work addresses the high computational complexity of Shapley value estimation in large-scale data valuation by modeling the utility function as a smooth functional of the mean embedding of empirical distributions in a reproducing kernel Hilbert space (RKHS). Leveraging tools from functional analysis and cooperative game theory, the authors conduct an asymptotic analysis that reveals how, as the number of data sources grows, the Shapley value is asymptotically characterized by a simple first-order dominant term. This finding elucidates the scaling behavior and structural properties of Shapley values in such settings. Building on this insight, the paper derives an interpretable and computationally tractable approximation, which not only provides a theoretical benchmark for existing estimation algorithms but also offers a principled foundation for efficient and reliable valuation in large-scale data markets.