# Barycentric Fused Gromov-Wasserstein Balancing for Causal Inference under Multiple Treatments

arXiv: https://arxiv.org/abs/2608.22024 · HTML: https://mlsanity.com/paper/2608.22024/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 53 references that are in the map.
- First posted on arXiv 22 Aug 2026.

## Builds on (most cited first, 15 of 15)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [Optimal Transport for structured data with application on graphs](https://mlsanity.com/paper/1805.09114/) (2018)
- [Fused Gromov-Wasserstein distance for structured objects: theoretical foundations and mathematical properties](https://mlsanity.com/paper/1811.02834/) (2018)
- [Learning Disentangled Representations for Counterfactual Regression via Mutual Information Minimization](https://mlsanity.com/paper/2206.01022/) (2022)
- [Locality Preserving Joint Transfer for Domain Adaptation](https://mlsanity.com/paper/1906.07441/) (2019)
- [Reliable Off-Policy Learning for Dosage Combinations](https://mlsanity.com/paper/2305.19742/) (2023)
- [Approximating 1-Wasserstein Distance with Trees](https://mlsanity.com/paper/2206.12116/) (2022)
- [Hierarchical Capsule Prediction Network for Marketing Campaigns Effect](https://mlsanity.com/paper/2208.10113/) (2022)
- [Unsupervised Multilingual Alignment using Wasserstein Barycenter](https://mlsanity.com/paper/2002.00743/) (2020)
- [How causal machine learning can leverage marketing strategies: Assessing and improving the performance of a coupon campaign](https://mlsanity.com/paper/2204.10820/) (2022)
- [Individualised Treatment Effects Estimation with Composite Treatments and Composite Outcomes](https://mlsanity.com/paper/2502.08282/) (2025)
- [Multiple Treatments Causal Effects Estimation with Task Embeddings and Balanced Representation Learning](https://mlsanity.com/paper/2511.09814/) (2025)

## Abstract (arXiv)

Estimating heterogeneous single and interaction treatment effects from observational data under multiple simultaneous treatments is crucial for decision-making. To mitigate estimation variance, previous studies balance representation distributions between every pair of treatment patterns. However, such pairwise balancing scales quadratically with the number of treatment patterns and fails to preserve consistent local proximity structures across patterns, which degrades counterfactual estimation. To address these challenges, we propose the Causal Inference for Heterogeneous Single and Interaction Treatment Effects Network (CIHSI-Net), a deep learning framework built on a novel Barycentric Fused Gromov-Wasserstein Balancing (BFG-WB) objective. BFG-WB aligns the representation distribution of each treatment pattern with a shared Wasserstein barycenter, achieving global alignment while reducing the computational complexity from quadratic to linear, and its Fused Gromov-Wasserstein discrepancy preserves the local proximity structures essential for reliable heterogeneous effect estimation. Simulation studies show that CIHSI-Net consistently outperforms state-of-the-art baselines, and an application to real-world marketing data demonstrates its practical utility in complex multi-treatment scenarios.