将单纯形数据映射到欧氏空间，实现类别分布的高效建模与采样。
Simplex-to-Euclidean Bijections for Categorical Flow Matching
- 用Aitchison几何定义光滑双射，把单纯形转为欧氏空间
- 通过狄利克雷插值将离散数据连续化，支持精确还原原始分布
- 无需黎曼几何或自定义噪声，性能媲美现有方法
我们提出一种在单纯形上学习和采样的方法。该方法通过平滑双射将开单纯形映射到欧氏空间，利用Aitchison几何定义映射关系，并采用狄利克雷插值将离散观测值去量化为连续表示，从而在欧氏空间中实现密度建模，同时仍能精确恢复原始离散分布。相较于以往在单纯形上使用黎曼几何或自定义噪声过程的方法，本方法在欧氏空间中运行且保持Aitchison几何结构，在合成数据和真实数据集上均达到可比性能。
原文摘要 · Abstract (English)
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.
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