Simplex-to-Euclidean Bijections for Categorical Flow Matching

Bernardo Williams, Victor M. Yeom-Song, Marcelo Hartmann, Arto Klami
Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, PMLR 300:280-288, 2026.

Abstract

We propose a method for learning and sampling from probability distributions supported on the simplex. Our approach maps the open simplex to Euclidean space via smooth bijections, leveraging the Aitchison geometry to define the mappings, and supports modeling categorical data by a Dirichlet interpolation that dequantizes discrete observations into continuous ones. This enables density modeling in Euclidean space through the bijection while still allowing exact recovery of the original discrete distribution. Compared to previous methods that operate on the simplex using Riemannian geometry or custom noise processes, our approach works in Euclidean space while respecting the Aitchison geometry, and achieves competitive performance on both synthetic and real-world data sets.

Cite this Paper

BibTeX
@InProceedings{pmlr-v300-williams26a, title = { Simplex-to-Euclidean Bijections for Categorical Flow Matching }, author = {Williams, Bernardo and Yeom-Song, Victor M. and Hartmann, Marcelo and Klami, Arto}, booktitle = {Proceedings of The 29th International Conference on Artificial Intelligence and Statistics}, pages = {280--288}, year = {2026}, editor = {Khan, Emtiyaz and Li, Yingzhen and Solin, Arno and Ramdas, Aaditya}, volume = {300}, series = {Proceedings of Machine Learning Research}, month = {02--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v300/main/assets/williams26a/williams26a.pdf}, url = {https://proceedings.mlr.press/v300/williams26a.html}, abstract = { We propose a method for learning and sampling from probability distributions supported on the simplex. Our approach maps the open simplex to Euclidean space via smooth bijections, leveraging the Aitchison geometry to define the mappings, and supports modeling categorical data by a Dirichlet interpolation that dequantizes discrete observations into continuous ones. This enables density modeling in Euclidean space through the bijection while still allowing exact recovery of the original discrete distribution. Compared to previous methods that operate on the simplex using Riemannian geometry or custom noise processes, our approach works in Euclidean space while respecting the Aitchison geometry, and achieves competitive performance on both synthetic and real-world data sets. } }
Endnote
%0 Conference Paper %T Simplex-to-Euclidean Bijections for Categorical Flow Matching %A Bernardo Williams %A Victor M. Yeom-Song %A Marcelo Hartmann %A Arto Klami %B Proceedings of The 29th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2026 %E Emtiyaz Khan %E Yingzhen Li %E Arno Solin %E Aaditya Ramdas %F pmlr-v300-williams26a %I PMLR %P 280--288 %U https://proceedings.mlr.press/v300/williams26a.html %V 300 %X We propose a method for learning and sampling from probability distributions supported on the simplex. Our approach maps the open simplex to Euclidean space via smooth bijections, leveraging the Aitchison geometry to define the mappings, and supports modeling categorical data by a Dirichlet interpolation that dequantizes discrete observations into continuous ones. This enables density modeling in Euclidean space through the bijection while still allowing exact recovery of the original discrete distribution. Compared to previous methods that operate on the simplex using Riemannian geometry or custom noise processes, our approach works in Euclidean space while respecting the Aitchison geometry, and achieves competitive performance on both synthetic and real-world data sets.
APA
Williams, B., Yeom-Song, V.M., Hartmann, M. & Klami, A.. (2026). Simplex-to-Euclidean Bijections for Categorical Flow Matching . Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 300:280-288 Available from https://proceedings.mlr.press/v300/williams26a.html.

Related Material