# The Representational Limit of Scalar Interactions: An Interventional Decomposition

arXiv: https://arxiv.org/abs/2606.19410 · HTML: https://mlsanity.com/paper/2606.19410/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 57 references that are in the map.
- First posted on arXiv 17 Jun 2026.

## Builds on (most cited first, 30 of 32)

- [Grad-CAM: Visual Explanations from Deep Networks via Gradient-based Localization](https://mlsanity.com/paper/1610.02391/) (2016)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [Locating and Editing Factual Associations in GPT](https://mlsanity.com/paper/2202.05262/) (2022)
- [Visualizing and Understanding Convolutional Networks](https://mlsanity.com/paper/1311.2901/) (2013)
- [Interpretability in the Wild: a Circuit for Indirect Object Identification in GPT-2 small](https://mlsanity.com/paper/2211.00593/) (2022)
- [ChestX-ray8: Hospital-scale Chest X-ray Database and Benchmarks on Weakly-Supervised Classification and Localization of Common Thorax Diseases](https://mlsanity.com/paper/1705.02315/) (2017)
- [RISE: Randomized Input Sampling for Explanation of Black-box Models](https://mlsanity.com/paper/1806.07421/) (2018)
- [Towards Automated Circuit Discovery for Mechanistic Interpretability](https://mlsanity.com/paper/2304.14997/) (2023)
- [Distribution-Free Predictive Inference For Regression](https://mlsanity.com/paper/1604.04173/) (2016)
- [Interpretable Explanations of Black Boxes by Meaningful Perturbation](https://mlsanity.com/paper/1704.03296/) (2017)
- [The many Shapley values for model explanation](https://mlsanity.com/paper/1908.08474/) (2019)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Quantus: An Explainable AI Toolkit for Responsible Evaluation of Neural Network Explanations and Beyond](https://mlsanity.com/paper/2202.06861/) (2022)
- [Feature relevance quantification in explainable AI: A causal problem](https://mlsanity.com/paper/1910.13413/) (2019)
- [The Shapley Taylor Interaction Index](https://mlsanity.com/paper/1902.05622/) (2019)
- [OpenXAI: Towards a Transparent Evaluation of Model Explanations](https://mlsanity.com/paper/2206.11104/) (2022)
- [Causal Shapley Values: Exploiting Causal Knowledge to Explain Individual Predictions of Complex Models](https://mlsanity.com/paper/2011.01625/) (2020)
- [Explaining Explanations: Axiomatic Feature Interactions for Deep Networks](https://mlsanity.com/paper/2002.04138/) (2020)
- [How does this interaction affect me? Interpretable attribution for feature interactions](https://mlsanity.com/paper/2006.10965/) (2020)
- [Faith-Shap: The Faithful Shapley Interaction Index](https://mlsanity.com/paper/2203.00870/) (2022)
- [shapiq: Shapley Interactions for Machine Learning](https://mlsanity.com/paper/2410.01649/) (2024)
- [KernelSHAP-IQ: Weighted Least-Square Optimization for Shapley Interactions](https://mlsanity.com/paper/2405.10852/) (2024)
- [A Novel Approach to the Partial Information Decomposition](https://mlsanity.com/paper/1908.08642/) (2019)
- [DiffusionPID: Interpreting Diffusion via Partial Information Decomposition](https://mlsanity.com/paper/2406.05191/) (2024)
- [Explaining Graph Neural Networks via Structure-aware Interaction Index](https://mlsanity.com/paper/2405.14352/) (2024)
- [Explaining Local, Global, And Higher-Order Interactions In Deep Learning](https://mlsanity.com/paper/2006.08601/) (2020)
- [SHAP Meets Tensor Networks: Provably Tractable Explanations with Parallelism](https://mlsanity.com/paper/2510.21599/) (2025)
- [Assessing high-order effects in feature importance via predictability decomposition](https://mlsanity.com/paper/2412.09964/) (2024)

## Built on by (most cited first, 1 of 1)

- [Conditional Co-Ablation: Recovering Self-Repair Backups in Transformer Circuits](https://mlsanity.com/paper/2607.01940/) (2026)

## Abstract (arXiv)

Signed pairwise interaction scores fundamentally conflate uniqueness (U), redundancy (R), and synergy (S). We prove this on a minimal 3-way XOR structural causal model: faithful indices such as Shapley-Taylor return zero per pair, whereas projective indices such as Shapley Interaction spread the third-order effect into pair scalars that conflate the three mechanisms. We introduce Stochastic Hi-Fi, a post-hoc, retraining-free predictability decomposition that estimates per-feature U/R/S profiles by interventional masked inference. The estimator provides exact interventional semantics, finite-sample Monte Carlo bounds, strict variance reduction from coupled diamond sampling, and uniform finite-vocabulary convergence. Across tabular SCMs, Stochastic Hi-Fi recovers structure missed by scalar baselines (up to 411x larger interaction-magnitude recovery ratios). It also separates redundant and synergistic heads in the GPT-2 IOI circuit. On NIH ChestX-ray14, Stochastic Hi-Fi matches GradCAM on Pointing Game and improves substantially on Deletion AUC.