Language Modeling with Hyperspherical Flows

• URL: http://arxiv.org/abs/2605.11125v2
• Date: Sat, 16 May 2026 19:59:59 GMT
• Title: Language Modeling with Hyperspherical Flows
• Abstract summary: Recent Flow Language Models (FLMs) apply continuous flows to language, transporting noise to data with a deterministic ODE.<n>$mathbbS$-FLM generates sequences by rotating vectors in $mathbbSd-1$ along a velocity field learned with cross-entropy.<n>$mathbbS$-FLM substantially improves continuous flow language models on large-vocabulary reasoning.
• Score: 12.083218729202963
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Discrete Diffusion Language Models progressed rapidly as an alternative to autoregressive (AR) models, motivated by their parallel generation abilities. However, for tractability, discrete diffusion models sample from a factorized distribution, which is less expressive than AR. Recent Flow Language Models (FLMs) apply continuous flows to language, transporting noise to data with a deterministic ODE that avoids factorized sampling. FLMs operate on one-hot vectors whose dimension scales with the vocabulary size, making FLMs costly to train. Moreover, since all distinct one-hot embeddings are equidistant in $\ell_2$, adding Gaussian noise does not have a clear semantic interpretation (unlike images, where Gaussian noise progressively degrades structure). We introduce $\mathbb{S}$-FLM, a latent FLM in the hypersphere. $\mathbb{S}$-FLM generates sequences by rotating vectors in $\mathbb{S}^{d-1}$ along a velocity field learned with cross-entropy, avoiding the overhead of materializing one-hot vectors. Previous FLMs match AR in Generative Perplexity (Gen.\ PPL), but samples with high likelihood are not necessarily correct in verifiable domains such as math and code. $\mathbb{S}$-FLM substantially improves continuous flow language models on large-vocabulary reasoning and closes the gap to masked diffusion under standard-temperature sampling ($T=1$), while a gap remains under optimized low-temperature ($T=0.1$) decoding.

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