# Counterfactual Explanations for Oblique Decision Trees: Exact, Efficient Algorithms

arXiv: https://arxiv.org/abs/2103.01096 · HTML: https://mlsanity.com/paper/2103.01096/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 53 references that are in the map.
- First posted on arXiv 1 Mar 2021.

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

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps](https://mlsanity.com/paper/1312.6034/) (2013)
- [Visualizing and Understanding Convolutional Networks](https://mlsanity.com/paper/1311.2901/) (2013)
- [Counterfactual Explanations without Opening the Black Box: Automated Decisions and the GDPR](https://mlsanity.com/paper/1711.00399/) (2017)
- [A Survey Of Methods For Explaining Black Box Models](https://mlsanity.com/paper/1802.01933/) (2018)
- [The Mythos of Model Interpretability](https://mlsanity.com/paper/1606.03490/) (2016)
- [European Union regulations on algorithmic decision-making and a "right to explanation"](https://mlsanity.com/paper/1606.08813/) (2016)
- [Actionable Recourse in Linear Classification](https://mlsanity.com/paper/1809.06514/) (2018)
- [Explanations based on the Missing: Towards Contrastive Explanations with Pertinent Negatives](https://mlsanity.com/paper/1802.07623/) (2018)
- [Interpretable Counterfactual Explanations Guided by Prototypes](https://mlsanity.com/paper/1907.02584/) (2019)
- [The What-If Tool: Interactive Probing of Machine Learning Models](https://mlsanity.com/paper/1907.04135/) (2019)
- [Model-Agnostic Counterfactual Explanations for Consequential Decisions](https://mlsanity.com/paper/1905.11190/) (2019)
- [Inverting Visual Representations with Convolutional Networks](https://mlsanity.com/paper/1506.02753/) (2015)
- [Efficient Search for Diverse Coherent Explanations](https://mlsanity.com/paper/1901.04909/) (2019)
- [Visualizing Deep Convolutional Neural Networks Using Natural Pre-Images](https://mlsanity.com/paper/1512.02017/) (2015)
- [Interpretable Predictions of Tree-based Ensembles via Actionable Feature Tweaking](https://mlsanity.com/paper/1706.06691/) (2017)
- [An Experimental Comparison of Old and New Decision Tree Algorithms](https://mlsanity.com/paper/1911.03054/) (2019)
- [Sampling the "Inverse Set" of a Neuron: An Approach to Understanding Neural Nets](https://mlsanity.com/paper/1910.04857/) (2019)

## Built on by (most cited first, 14 of 14)

- [Data-centric Artificial Intelligence: A Survey](https://mlsanity.com/paper/2303.10158/) (2023)
- [Efficient XAI Techniques: A Taxonomic Survey](https://mlsanity.com/paper/2302.03225/) (2023)
- [Achieving Diversity in Counterfactual Explanations: a Review and Discussion](https://mlsanity.com/paper/2305.05840/) (2023)
- [GLANCE: Global Actions in a Nutshell for Counterfactual Explainability](https://mlsanity.com/paper/2405.18921/) (2024)
- [Very fast, approximate counterfactual explanations for decision forests](https://mlsanity.com/paper/2303.02883/) (2023)
- [An exact counterfactual-example-based approach to tree-ensemble models interpretability](https://mlsanity.com/paper/2105.14820/) (2021)
- [Sparse Oblique Decision Trees: A Tool to Understand and Manipulate Neural Net Features](https://mlsanity.com/paper/2104.02922/) (2021)
- [counterfactuals: An R Package for Counterfactual Explanation Methods](https://mlsanity.com/paper/2304.06569/) (2023)
- [Towards Non-Adversarial Algorithmic Recourse](https://mlsanity.com/paper/2403.10330/) (2024)
- [DiCoFlex: Model-agnostic diverse counterfactuals with flexible control](https://mlsanity.com/paper/2505.23700/) (2025)
- [Explainable Reinforcement Learning for Formula One Race Strategy](https://mlsanity.com/paper/2501.04068/) (2025)
- [Local and Regional Counterfactual Rules: Summarized and Robust Recourses](https://mlsanity.com/paper/2209.14568/) (2022)
- [Learning Inter-Annual Flood Loss Risk Models From Historical Flood Insurance Claims and Extreme Rainfall Data](https://mlsanity.com/paper/2212.08660/) (2022)
- [Inverse classification with logistic and softmax classifiers: efficient optimization](https://mlsanity.com/paper/2309.08945/) (2023)

## Abstract (arXiv)

We consider counterfactual explanations, the problem of minimally adjusting features in a source input instance so that it is classified as a target class under a given classifier. This has become a topic of recent interest as a way to query a trained model and suggest possible actions to overturn its decision. Mathematically, the problem is formally equivalent to that of finding adversarial examples, which also has attracted significant attention recently. Most work on either counterfactual explanations or adversarial examples has focused on differentiable classifiers, such as neural nets. We focus on classification trees, both axis-aligned and oblique (having hyperplane splits). Although here the counterfactual optimization problem is nonconvex and nondifferentiable, we show that an exact solution can be computed very efficiently, even with high-dimensional feature vectors and with both continuous and categorical features, and demonstrate it in different datasets and settings. The results are particularly relevant for finance, medicine or legal applications, where interpretability and counterfactual explanations are particularly important.