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

TL;DR

TT-FSI introduces a scalable, exact method to compute Faithful Shapley Interactions (FSI) by representing the FSI operator as a Matrix Product Operator with TT-rank $O(\ell d)$. It decomposes the closed-form solution into a rank-1 Möbius transform and a two-counter, polynomial-rank correction term, enabling an efficient left-to-right sweep with time $O(\ell^2 d^3 \cdot 2^d)$ and core storage $O(\ell d^2)$. Empirically, TT-FSI delivers up to 280x speedups and 290x memory reductions on datasets with $d$ from 8 to 20, successfully handling $d=20$ where competing methods fail due to memory constraints. The approach is model-agnostic and leverages tensor-network techniques to make faithful interaction explanations practical for moderate-dimensional tabular data; future work includes integrating with sampling-based value estimation and exploring GPU acceleration.

Abstract

The Faithful Shapley Interaction (FSI) index uniquely satisfies the faithfulness axiom among Shapley interaction indices, but computing FSI requires $O(d^\ell \cdot 2^d)$ 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 $O(\ell d)$, enabling an efficient sweep algorithm with $O(\ell^2 d^3 \cdot 2^d)$ time and $O(\ell d^2)$ 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.

Figures (3)

• Figure 1: Finite-state machine for correction operator ($d=3$, $\ell=2$). States $(s,t)$ track cumulative counts. Colors: blue ($\notin S, \notin T$), green ($\notin S, \in T$), red ($\in S \cap T$). State count grows polynomially: $1 \to 3 \to 6 \to 9$.
• Figure 2: Memory comparison ($\ell=3$). (a) Peak memory on log scale. Baseline is most memory-efficient but too slow for $d \geq 16$. SHAP-IQ exhibits $O(4^d)$ memory growth. (b) Memory ratio vs TT-FSI.
• Figure 3: TT-FSI vs SHAP-IQ pairwise interaction scores on California Housing. Perfect agreement (correlation $> 0.9999$).

Theorems & Definitions (15)

• Lemma 1: TT-rank upper bound
• Lemma 2: Kronecker product has TT-rank 1 oseledets2011tensor
• proof
• Lemma 3: TT-rank upper bound, restated
• proof
• Lemma 4: Weight function correctness
• proof
• Lemma 5: Time complexity
• proof
• Lemma 6: Precontraction preserves TT-rank