# Unifying local and global model explanations by functional decomposition of low dimensional structures

arXiv: https://arxiv.org/abs/2208.06151 · HTML: https://mlsanity.com/paper/2208.06151/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 49 references that are in the map.
- First posted on arXiv 12 Aug 2022.

## Builds on (most cited first, 19 of 19)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [Counterfactual Fairness](https://mlsanity.com/paper/1703.06856/) (2017)
- [Fooling LIME and SHAP: Adversarial Attacks on Post hoc Explanation Methods](https://mlsanity.com/paper/1911.02508/) (2019)
- [The many Shapley values for model explanation](https://mlsanity.com/paper/1908.08474/) (2019)
- [Problems with Shapley-value-based explanations as feature importance measures](https://mlsanity.com/paper/2002.11097/) (2020)
- [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)
- [True to the Model or True to the Data?](https://mlsanity.com/paper/2006.16234/) (2020)
- [Asymmetric Shapley values: incorporating causal knowledge into model-agnostic explainability](https://mlsanity.com/paper/1910.06358/) (2019)
- [Faith-Shap: The Faithful Shapley Interaction Index](https://mlsanity.com/paper/2203.00870/) (2022)
- [Efficient nonparametric statistical inference on population feature importance using Shapley values](https://mlsanity.com/paper/2006.09481/) (2020)
- [Visualizing the Feature Importance for Black Box Models](https://mlsanity.com/paper/1804.06620/) (2018)
- [Purifying Interaction Effects with the Functional ANOVA: An Efficient Algorithm for Recovering Identifiable Additive Models](https://mlsanity.com/paper/1911.04974/) (2019)
- [Joint Shapley values: a measure of joint feature importance](https://mlsanity.com/paper/2107.11357/) (2021)
- [Feature Synergy, Redundancy, and Independence in Global Model Explanations using SHAP Vector Decomposition](https://mlsanity.com/paper/2107.12436/) (2021)
- [Threading the Needle of On and Off-Manifold Value Functions for Shapley Explanations](https://mlsanity.com/paper/2202.11919/) (2022)
- [Random Planted Forest: a directly interpretable tree ensemble](https://mlsanity.com/paper/2012.14563/) (2020)

## Built on by (most cited first, 18 of 18)

- [From Shapley Values to Generalized Additive Models and back](https://mlsanity.com/paper/2209.04012/) (2022)
- [KernelSHAP-IQ: Weighted Least-Square Optimization for Shapley Interactions](https://mlsanity.com/paper/2405.10852/) (2024)
- [Exact Computation of Any-Order Shapley Interactions for Graph Neural Networks](https://mlsanity.com/paper/2501.16944/) (2025)
- [Statistical Aspects of SHAP: Functional ANOVA for Model Interpretation](https://mlsanity.com/paper/2208.09970/) (2022)
- [A Guide to Feature Importance Methods for Scientific Inference](https://mlsanity.com/paper/2404.12862/) (2024)
- [Unifying Feature-Based Explanations with Functional ANOVA and Cooperative Game Theory](https://mlsanity.com/paper/2412.17152/) (2024)
- [Tree Ensemble Explainability through the Hoeffding Functional Decomposition and TreeHFD Algorithm](https://mlsanity.com/paper/2510.24815/) (2025)
- [Why You Should Not Trust Interpretations in Machine Learning: Adversarial Attacks on Partial Dependence Plots](https://mlsanity.com/paper/2404.18702/) (2024)
- [Interpretable Machine Learning for Survival Analysis](https://mlsanity.com/paper/2403.10250/) (2024)
- [Fast Estimation of Partial Dependence Functions using Trees](https://mlsanity.com/paper/2410.13448/) (2024)
- [Disentangling Interactions and Dependencies in Feature Attribution](https://mlsanity.com/paper/2410.23772/) (2024)
- [Shapley Curves: A Smoothing Perspective](https://mlsanity.com/paper/2211.13289/) (2022)
- [STRIDE: Subset-Free Functional Decomposition for XAI in Tabular Settings](https://mlsanity.com/paper/2509.09070/) (2025)
- [Conditional expectation network for SHAP](https://mlsanity.com/paper/2307.10654/) (2023)
- [Explainable artificial intelligence model for identifying Market Value in Professional Soccer Players](https://mlsanity.com/paper/2311.04599/) (2023)
- [midr: Learning from Black-Box Models by Maximum Interpretation Decomposition](https://mlsanity.com/paper/2506.08338/) (2025)
- [Woodelf++: A Fast and Unified Partial Dependence Plot Algorithm for Decision Tree Ensembles](https://mlsanity.com/paper/2605.14578/) (2026)
- [Beyond Additive Decompositions: Interpretability Through Separability](https://mlsanity.com/paper/2605.31200/) (2026)

## Abstract (arXiv)

We consider a global representation of a regression or classification function by decomposing it into the sum of main and interaction components of arbitrary order. We propose a new identification constraint that allows for the extraction of interventional SHAP values and partial dependence plots, thereby unifying local and global explanations. With our proposed identification, a feature's partial dependence plot corresponds to the main effect term plus the intercept. The interventional SHAP value of feature k is a weighted sum of the main component and all interaction components that include k, with the weights given by the reciprocal of the component's dimension. This brings a new perspective to local explanations such as SHAP values which were previously motivated by game theory only. We show that the decomposition can be used to reduce direct and indirect bias by removing all components that include a protected feature. Lastly, we motivate a new measure of feature importance. In principle, our proposed functional decomposition can be applied to any machine learning model, but exact calculation is only feasible for low-dimensional structures or ensembles of those. We provide an algorithm and efficient implementation for gradient-boosted trees (xgboost) and random planted forest. Conducted experiments suggest that our method provides meaningful explanations and reveals interactions of higher orders. The proposed methods are implemented in an R package, available at urlhttps://github.com/PlantedML/glex.