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Tractable Shapley Values and Interactions via Tensor Networks

Farzaneh Heidari, Chao Li, Guillaume Rabusseau

TL;DR

TN-SHAP introduces a tensor-network surrogate to compute exact Shapley values and Shapley-style interactions with only $O(n)$ forward evaluations, by exploiting multilinearity and using diagonal selectors to aggregate coalitions by size. A polynomial interpolation step on a univariate probe $G_i(t;x)$ recovers size-grouped marginal contributions, while TN contractions with bond dimension $oldsymbol{hi}$ keep the computation tractable as $O(n ext{poly}(oldsymbol{hi})+n^2)$. The framework supports feature-map lifts to preserve multilinearity and improve surrogate fidelity, achieving significant speedups (e.g., $25$–$1000 imes$) over KernelSHAP-IQ with comparable accuracy on UCI benchmarks, and providing theoretical guarantees on approximation error and runtime. Empirically, TN-SHAP recovers first-, second-, and third-order interactions on synthetic multilinear targets and demonstrates robust performance across real-world models, while offering diagnostics through rank and fidelity analyses. Overall, TN-SHAP bridges local interpretability with scalable, faithful explanations by turning exponential coalition evaluation into structured TN contractions and interpolation, enabling tractable, reliable Shapley-based explanations in higher dimensions.

Abstract

We show how to replace the O(2^n) coalition enumeration over n features behind Shapley values and Shapley-style interaction indices with a few-evaluation scheme on a tensor-network (TN) surrogate: TN-SHAP. The key idea is to represent a predictor's local behavior as a factorized multilinear map, so that coalitional quantities become linear probes of a coefficient tensor. TN-SHAP replaces exhaustive coalition sweeps with just a small number of targeted evaluations to extract order-k Shapley interactions. In particular, both order-1 (single-feature) and order-2 (pairwise) computations have cost O(n*poly(chi) + n^2), where chi is the TN's maximal cut rank. We provide theoretical guarantees on the approximation error and tractability of TN-SHAP. On UCI datasets, our method matches enumeration on the fitted surrogate while reducing evaluation by orders of magnitude and achieves 25-1000x wall-clock speedups over KernelSHAP-IQ at comparable accuracy, while amortizing training across local cohorts.

Tractable Shapley Values and Interactions via Tensor Networks

TL;DR

TN-SHAP introduces a tensor-network surrogate to compute exact Shapley values and Shapley-style interactions with only forward evaluations, by exploiting multilinearity and using diagonal selectors to aggregate coalitions by size. A polynomial interpolation step on a univariate probe recovers size-grouped marginal contributions, while TN contractions with bond dimension keep the computation tractable as . The framework supports feature-map lifts to preserve multilinearity and improve surrogate fidelity, achieving significant speedups (e.g., –) over KernelSHAP-IQ with comparable accuracy on UCI benchmarks, and providing theoretical guarantees on approximation error and runtime. Empirically, TN-SHAP recovers first-, second-, and third-order interactions on synthetic multilinear targets and demonstrates robust performance across real-world models, while offering diagnostics through rank and fidelity analyses. Overall, TN-SHAP bridges local interpretability with scalable, faithful explanations by turning exponential coalition evaluation into structured TN contractions and interpolation, enabling tractable, reliable Shapley-based explanations in higher dimensions.

Abstract

We show how to replace the O(2^n) coalition enumeration over n features behind Shapley values and Shapley-style interaction indices with a few-evaluation scheme on a tensor-network (TN) surrogate: TN-SHAP. The key idea is to represent a predictor's local behavior as a factorized multilinear map, so that coalitional quantities become linear probes of a coefficient tensor. TN-SHAP replaces exhaustive coalition sweeps with just a small number of targeted evaluations to extract order-k Shapley interactions. In particular, both order-1 (single-feature) and order-2 (pairwise) computations have cost O(n*poly(chi) + n^2), where chi is the TN's maximal cut rank. We provide theoretical guarantees on the approximation error and tractability of TN-SHAP. On UCI datasets, our method matches enumeration on the fitted surrogate while reducing evaluation by orders of magnitude and achieves 25-1000x wall-clock speedups over KernelSHAP-IQ at comparable accuracy, while amortizing training across local cohorts.
Paper Structure (67 sections, 2 theorems, 40 equations, 5 figures, 10 tables, 3 algorithms)

This paper contains 67 sections, 2 theorems, 40 equations, 5 figures, 10 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background and Preliminaries
  3. Coalitional Values and the Shapley Value
  4. Tractable Shapley values with multilinearity
  5. Tensors and Tensor Networks
  6. Tensors as multilinear maps
  7. Tensor Networks (TN)
  8. Contraction (forward pass).
  9. TN Surrogates
  10. Method
  11. Multilinearity & Efficient Computation
  12. Computing Shapley Values via Diagonal Selectors
  13. Diagonal Selectors
  14. Polynomial interpolation
  15. Efficient Implementation with Tensor Networks
  16. ...and 52 more sections

Key Result

Theorem 4.1

Let $f$ be the original model and $g$ a multilinear surrogate. If the surrogate approximates coalition values uniformly well: then any size-weighted coalitional index $\Phi_i = \sum_{C\subseteq N\setminus\{i\}} w(|C|,n)\,[v(x, C\cup\{i\})-v(x, C)]$ satisfies:

Figures (5)

  • Figure 1: Two common tensor-network topologies used in this work: a tensor train (left) and a balanced binary tree (right). Only physical input legs $\vec{x}_k$ are shown.
  • Figure 2: Two TNs with selector matrices $S_k(t)$: tensor train (left) and balanced binary tree (right).
  • Figure 3: Concrete: runtime (log $y$) vs. cosine ($x$); $k{=}2$ (left), $k{=}3$ (right). TN-SHAP achieves higher similarity at millisecond scale.
  • Figure 4: Training dynamics: $R^2$ vs. epoch for the student fit (Train $R^2$) and SII ($k{=}1,2,3$).
  • Figure 5: Rank ablation heatmap: safe-$R^2$ for student ranks under a ground-truth tensor network of rank 14.

Theorems & Definitions (4)

  • Theorem 4.1: Approximation Error for Coalitional Indices
  • Theorem 4.2: Tractability: From Exponential to Polynomial
  • proof : Proof of Theorem \ref{['thm:approximation']}
  • proof : Proof of Theorem \ref{['thm:tractability']}