Spherical Flows for Sampling Categorical Data

• URL: http://arxiv.org/abs/2605.05629v2
• Date: Fri, 08 May 2026 21:05:22 GMT
• Title: Spherical Flows for Sampling Categorical Data
• Abstract summary: We study the problem of learning generative models for discrete sequences in a continuous embedding space.<n>Experiments compare the vMF path against geodesic and Euclidean alternatives.<n>The combination of vMF and PC sampling significantly improves results on Sudoku and language modeling.
• Score: 6.88204255655161
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: We study the problem of learning generative models for discrete sequences in a continuous embedding space. Whereas prior approaches typically operate in Euclidean space or on the probability simplex, we instead work on the sphere $\mathbb S^{d-1}$. There the von Mises-Fisher (vMF) distribution induces a natural noise process and admits a closed-form conditional score. The conditional velocity is in general intractable. Exploiting the radial symmetry of the vMF density we reduce the continuity equation on $\mathbb S^{d-1}$ to a scalar ODE in the cosine similarity, whose unique bounded solution determines the velocity. The marginal velocity and marginal score on $(\mathbb S^{d-1})^L$ both decompose into posterior-weighted tangent sums that differ only by per-token scalar weights. This gives access to both ODE and predictor-corrector (PC) sampling. The posterior is the only learned object, trained by a cross-entropy loss. Experiments compare the vMF path against geodesic and Euclidean alternatives. The combination of vMF and PC sampling significantly improves results on Sudoku and language modeling.

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