Simplex-to-Euclidean Bijections for Categorical Flow Matching

TL;DR

This paper introduces a novel method for modeling categorical distributions by mapping the simplex to Euclidean space using Aitchison geometry, enabling effective density estimation and sampling while preserving the original discrete distribution.

Contribution

The paper presents a new bijective mapping from the simplex to Euclidean space based on Aitchison geometry, facilitating density modeling of categorical data with improved flexibility and performance.

Findings

Achieves competitive results on synthetic data

Supports exact recovery of original distributions

Works effectively on real-world categorical datasets

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.