Amortized Linear-time Exact Shapley Value for Product-Kernel Methods

Kernel methods are widely used in machine learning and statistics for their flexibility and expressive power, yet their black-box nature limits adoption in high-stakes applications. Shapley value--based attribution methods such as SHAP, and kernel-specific adaptations including RKHS-SHAP, provide a principled framework for explainability---but exact computation of Shapley values is generally intractable, forcing existing approaches to rely on approximations that incur unavoidable estimation error. We introduce PKeX-Shapley, an algorithm that exploits the multiplicative structure of product kernels to compute exact Shapley values for all $d$ features in quadratic time in $d$. The method rests on a distribution-free removal operator intrinsic to the product-kernel structure: removing a feature replaces its kernel factor with the multiplicative identity. This yields a parameter-free value function---requiring no sampling and no density estimation---and uniquely determines a functional decomposition of the model. Building on this value function, we develop shared recursive formulations that evaluate all feature attributions jointly, achieving amortized linear time per feature with numerical stability. Beyond predictive modeling, the framework extends to widely used kernel-based discrepancies such as the Maximum Mean Discrepancy (MMD) and the Hilbert--Schmidt Independence Criterion (HSIC), providing new tools for interpretable statistical analysis.