# Synthesising Counterfactual Explanations via Label-Conditional Gaussian Mixture Variational Autoencoders

arXiv: https://arxiv.org/abs/2510.04855 · HTML: https://mlsanity.com/paper/2510.04855/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 55 references that are in the map.
- First posted on arXiv 6 Oct 2025.

## Builds on (most cited first, 30 of 36)

- [Counterfactual Explanations without Opening the Black Box: Automated Decisions and the GDPR](https://mlsanity.com/paper/1711.00399/) (2017)
- [Unsupervised Deep Embedding for Clustering Analysis](https://mlsanity.com/paper/1511.06335/) (2015)
- [Explaining Machine Learning Classifiers through Diverse Counterfactual Explanations](https://mlsanity.com/paper/1905.07697/) (2019)
- [Semi-Supervised Learning with Deep Generative Models](https://mlsanity.com/paper/1406.5298/) (2014)
- [FACE: Feasible and Actionable Counterfactual Explanations](https://mlsanity.com/paper/1909.09369/) (2019)
- [Deep Unsupervised Clustering with Gaussian Mixture Variational Autoencoders](https://mlsanity.com/paper/1611.02648/) (2016)
- [Deep Structural Causal Models for Tractable Counterfactual Inference](https://mlsanity.com/paper/2006.06485/) (2020)
- [Multi-Objective Counterfactual Explanations](https://mlsanity.com/paper/2004.11165/) (2020)
- [Learning Model-Agnostic Counterfactual Explanations for Tabular Data](https://mlsanity.com/paper/1910.09398/) (2019)
- [Preserving Causal Constraints in Counterfactual Explanations for Machine Learning Classifiers](https://mlsanity.com/paper/1912.03277/) (2019)
- [Algorithmic recourse under imperfect causal knowledge: a probabilistic approach](https://mlsanity.com/paper/2006.06831/) (2020)
- [Towards Robust and Reliable Algorithmic Recourse](https://mlsanity.com/paper/2102.13620/) (2021)
- [CARLA: A Python Library to Benchmark Algorithmic Recourse and Counterfactual Explanation Algorithms](https://mlsanity.com/paper/2108.00783/) (2021)
- [Interpretable Predictions of Tree-based Ensembles via Actionable Feature Tweaking](https://mlsanity.com/paper/1706.06691/) (2017)
- [On Counterfactual Explanations under Predictive Multiplicity](https://mlsanity.com/paper/2006.13132/) (2020)
- [Weakly Supervised Disentangled Generative Causal Representation Learning](https://mlsanity.com/paper/2010.02637/) (2020)
- [NICE: An Algorithm for Nearest Instance Counterfactual Explanations](https://mlsanity.com/paper/2104.07411/) (2021)
- [On the Adversarial Robustness of Causal Algorithmic Recourse](https://mlsanity.com/paper/2112.11313/) (2021)
- [High Fidelity Image Counterfactuals with Probabilistic Causal Models](https://mlsanity.com/paper/2306.15764/) (2023)
- [Robust Counterfactual Explanations for Tree-Based Ensembles](https://mlsanity.com/paper/2207.02739/) (2022)
- [Evaluating Robustness of Counterfactual Explanations](https://mlsanity.com/paper/2103.02354/) (2021)
- [A Survey on the Robustness of Feature Importance and Counterfactual Explanations](https://mlsanity.com/paper/2111.00358/) (2021)
- [Measuring axiomatic soundness of counterfactual image models](https://mlsanity.com/paper/2303.01274/) (2023)
- [Robust Counterfactual Explanations in Machine Learning: A Survey](https://mlsanity.com/paper/2402.01928/) (2024)
- [Formalising the Robustness of Counterfactual Explanations for Neural Networks](https://mlsanity.com/paper/2208.14878/) (2022)
- [Scaling Guarantees for Nearest Counterfactual Explanations](https://mlsanity.com/paper/2010.04965/) (2020)
- [Beyond the convexity assumption: Realistic tabular data generation under quantifier-free real linear constraints](https://mlsanity.com/paper/2502.18237/) (2025)
- [Promoting Counterfactual Robustness through Diversity](https://mlsanity.com/paper/2312.06564/) (2023)
- [Recourse under Model Multiplicity via Argumentative Ensembling (Technical Report)](https://mlsanity.com/paper/2312.15097/) (2023)
- [A Survey on Deep Learning Approaches for Tabular Data Generation: Utility, Alignment, Fidelity, Privacy, Diversity, and Beyond](https://mlsanity.com/paper/2503.05954/) (2025)

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

- [FCA-Guided Counterfactual Explanations for Multi-Modal Breast Cancer Diagnosis: A Framework Achieving Perfect Validity with Emergent Sparsity](https://mlsanity.com/paper/2609.20067/) (2026)

## Abstract (arXiv)

Counterfactual explanations (CEs) provide recourse recommendations for individuals affected by algorithmic decisions. A key challenge is generating CEs that are robust against various perturbation types (e.g. input and model perturbations) while simultaneously satisfying other desirable properties. These include plausibility, ensuring CEs reside on the data manifold, and diversity, providing multiple distinct recourse options for single inputs. Existing methods, however, mostly struggle to address these multifaceted requirements in a unified, model-agnostic manner. We address these limitations by proposing a novel generative framework. First, we introduce the Label-conditional Gaussian Mixture Variational Autoencoder (L-GMVAE), a model trained to learn a structured latent space where each class label is represented by a set of Gaussian components with diverse, prototypical centroids. Building on this, we present LAPACE (LAtent PAth Counterfactual Explanations), a model-agnostic algorithm that synthesises entire paths of CE points by interpolating from inputs' latent representations to those learned latent centroids. This approach inherently ensures robustness to input changes, as all paths for a given target class converge to the same fixed centroids. Furthermore, the generated paths provide a spectrum of recourse options, allowing users to navigate the trade-off between proximity and plausibility while also encouraging robustness against model changes. In addition, user-specified actionability constraints can also be easily incorporated via lightweight gradient optimisation through the L-GMVAE's decoder. Comprehensive experiments show that LAPACE is computationally efficient and achieves competitive performance across eight quantitative metrics.