# GRALIS: Fusing Coalition and Gradient Attribution with Closed-Form Conservation Error and Finite-Sample Guarantees

arXiv: https://arxiv.org/abs/2605.05480 · HTML: https://mlsanity.com/paper/2605.05480/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 52 references that are in the map.
- First posted on arXiv 6 May 2026.

## Builds on (most cited first, 27 of 27)

- [Grad-CAM: Visual Explanations from Deep Networks via Gradient-based Localization](https://mlsanity.com/paper/1610.02391/) (2016)
- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps](https://mlsanity.com/paper/1312.6034/) (2013)
- [Grad-CAM++: Improved Visual Explanations for Deep Convolutional Networks](https://mlsanity.com/paper/1710.11063/) (2017)
- [RISE: Randomized Input Sampling for Explanation of Black-box Models](https://mlsanity.com/paper/1806.07421/) (2018)
- [Score-CAM: Score-Weighted Visual Explanations for Convolutional Neural Networks](https://mlsanity.com/paper/1910.01279/) (2019)
- [A Benchmark for Interpretability Methods in Deep Neural Networks](https://mlsanity.com/paper/1806.10758/) (2018)
- [The (Un)reliability of saliency methods](https://mlsanity.com/paper/1711.00867/) (2017)
- [Explaining NonLinear Classification Decisions with Deep Taylor Decomposition](https://mlsanity.com/paper/1512.02479/) (2015)
- [Explaining by Removing: A Unified Framework for Model Explanation](https://mlsanity.com/paper/2011.14878/) (2020)
- [Full-Gradient Representation for Neural Network Visualization](https://mlsanity.com/paper/1905.00780/) (2019)
- [Evaluating and Aggregating Feature-based Model Explanations](https://mlsanity.com/paper/2005.00631/) (2020)
- [Axiom-based Grad-CAM: Towards Accurate Visualization and Explanation of CNNs](https://mlsanity.com/paper/2008.02312/) (2020)
- [FastSHAP: Real-Time Shapley Value Estimation](https://mlsanity.com/paper/2107.07436/) (2021)
- [Use HiResCAM instead of Grad-CAM for faithful explanations of convolutional neural networks](https://mlsanity.com/paper/2011.08891/) (2020)
- [Which Explanation Should I Choose? A Function Approximation Perspective to Characterizing Post Hoc Explanations](https://mlsanity.com/paper/2206.01254/) (2022)
- [A Rigorous Study of Integrated Gradients Method and Extensions to Internal Neuron Attributions](https://mlsanity.com/paper/2202.11912/) (2022)
- [RKHS-SHAP: Shapley Values for Kernel Methods](https://mlsanity.com/paper/2110.09167/) (2021)
- [Evaluating Feature Attribution Methods in the Image Domain](https://mlsanity.com/paper/2202.12270/) (2022)
- [Axiomatic Explanations for Visual Search, Retrieval, and Similarity Learning](https://mlsanity.com/paper/2103.00370/) (2021)
- [Four Axiomatic Characterizations of the Integrated Gradients Attribution Method](https://mlsanity.com/paper/2306.13753/) (2023)
- [Unifying Attribution-Based Explanations Using Functional Decomposition](https://mlsanity.com/paper/2412.13623/) (2024)
- [An exploration of the influence of path choice in game-theoretic attribution algorithms](https://mlsanity.com/paper/2007.04169/) (2020)
- [Feature Attribution from First Principles](https://mlsanity.com/paper/2505.24729/) (2025)
- [Aumann-SHAP: The Geometry of Counterfactual Interaction Explanations in Machine Learning](https://mlsanity.com/paper/2603.14014/) (2026)

## Abstract (arXiv)

The main post-hoc XAI methods for deep networks -- GradCAM, SHAP, LIME, Integrated Gradients -- originate from heterogeneous theoretical foundations and are not naturally comparable within a single representation. A recent benchmark also finds their coalition-based members (GradCAM, KernelSHAP, LIME) and gradient-based members (Integrated Gradients and variants) empirically complementary, each outperforming the other on different faithfulness metrics, with method selection as the only proposed remedy (Gevaert et al., 2022). This work presents GRALIS (Gradient-Riesz Averaged Locally-Integrated Shapley), which fuses these two mechanisms -- a Shapley coalition weight and locality kernel, and a continuous Integrated-Gradients-style conditioned path -- into a single estimator, and equips it with two certified guarantees neither mechanism supplies alone: an exact, closed-form completeness deficit (an order-d interaction is attributed at a factor 1/d of its true value under Shapley weights and a multilinear F) and a finite-sample bound, O(1/sqrt(m)) + O(1/k^2), for the actual self-normalized ratio the algorithm returns. This fusion is underpinned by a representation-theoretic result: every additive, linear, continuous attribution functional admits a unique canonical representation via the Riesz Representation Theorem, proved componentwise (feature by feature) rather than as one form shared across features or methods. This class includes SHAP, IG and LIME, but not nonlinear functionals such as standard GradCAM or attention maps. Seven theorems further establish an exact correspondence with Shapley Interaction Values, affine-regime correspondences with the Hoeffding/Sobol decomposition, and a minimum-variance multi-scale extension. A preliminary experimental illustration on breast histology imaging is included; extended validation is in a companion paper (Fanale, 2026).