Computing Exact Shapley Values in Polynomial Time for Product-Kernel Methods

Kernel methods are widely used in machine learning due to their flexibility and expressive power. However, their black-box nature poses significant challenges to interpretability, limiting their adoption in high-stakes applications. Shapley value-based feature attribution techniques, such as SHAP and kernel-specific variants like RKHS-SHAP, offer a promising path toward explainability. Yet, computing exact Shapley values remains computationally intractable in general, motivating the development of various approximation schemes. In this work, we introduce PKeX-Shapley, a novel algorithm that utilizes the multiplicative structure of product kernels to enable the exact computation of Shapley values in polynomial time. We show that product-kernel models admit a functional decomposition that allows for a recursive formulation of Shapley values. This decomposition not only yields computational efficiency but also enhances interpretability in kernel-based learning. We also demonstrate how our framework can be generalized to explain kernel-based statistical discrepancies such as the Maximum Mean Discrepancy (MMD) and the Hilbert-Schmidt Independence Criterion (HSIC), thus offering new tools for interpretable statistical inference.