Flow Duality and Source Geometry for Categorical Generation
Abstract
Continuous and discrete flow matching are usually treated as separate constructions. This paper identifies a duality between them: projecting continuous convex-interpolant flows with one-hot targets through a position-wise argmax yields discrete convex-interpolant flows. The result requires source laws with appropriate coordinate symmetry and regularity, and it makes the continuous source distribution an explicit design choice for categorical generation. We derive the induced discrete interpolation behavior for Gaussian, bounded-uniform, and centered negative-exponential sources, showing that different source geometries lead to qualitatively different transition timing and vocabulary-size dependence. Small visual diagnostics and a short language-modeling pilot suggest that these source-design effects can also appear in learned transports and early generative quality.
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BibTeX
@misc{haxholli2026flow,
title = {{Flow Duality and Source Geometry for Categorical Generation}},
author = {Etrit Haxholli},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10863},
eprint = {2609.10863},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10863},
}
RIS
TY - UNPB
TI - Flow Duality and Source Geometry for Categorical Generation
AU - Etrit Haxholli
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.10863
UR - https://arxiv.org/abs/2609.10863
AB - Continuous and discrete flow matching are usually treated as separate constructions. This paper identifies a duality between them: projecting continuous convex-interpolant flows with one-hot targets through a position-wise argmax yields discrete convex-interpolant flows. The result requires source laws with appropriate coordinate symmetry and regularity, and it makes the continuous source distribution an explicit design choice for categorical generation. We derive the induced discrete interpolation behavior for Gaussian, bounded-uniform, and centered negative-exponential sources, showing that different source geometries lead to qualitatively different transition timing and vocabulary-size dependence. Small visual diagnostics and a short language-modeling pilot suggest that these source-design effects can also appear in learned transports and early generative quality.
ER -