TT-FSI: Scalable Faithful Shapley Interactions via Tensor-Train

The Faithful Shapley Interaction (FSI) index uniquely satisfies the faithfulness axiom among Shapley interaction indices, but computing FSI requires $O(d^{\mathrm{\ell }}\cdot 2^d)$ time and existing implementations use $O(4^d)$ memory. We present TT-FSI, which exploits FSI’s algebraic structure via Matrix Product Operators (MPO). Our main theoretical contribution is proving that the linear operator $v\mapsto \text{FSI}(v)$ admits an MPO representation with TT-rank $O(\mathrm{\ell }d)$, enabling an efficient sweep algorithm with $O({\mathrm{\ell }}^2{d}^3\cdot 2^d)$ time and $O(\mathrm{\ell }{d}^2)$ core storage an exponential improvement over existing methods. Experiments on six datasets ($d=8$ to $d=20$) demonstrate up to 280$\times$ speedup over baseline, 85$\times$ over SHAP-IQ, and 290$\times$ memory reduction. TT-FSI scales to $d=20$ (1M coalitions) where all competing methods fail.