# Spherical Brownian Diffusion for Language Modeling

Authors: Andrea Miele, Ilija Bogunovic.

Corrupts token embeddings with Brownian motion on a sphere and trains a model to recover clean tokens. Controlled comparisons separate the effect of the corruption process from the sampling method on Sudoku, math problems and text generation.

First public: 2026-10-26.
ArXiv initial submission: 2026-10-26.

[Original paper](https://arxiv.org/abs/2610.35101) · [PDF](https://arxiv.org/pdf/2610.35101) · [HTML](https://arxiv.org/html/2610.35101v1) · [Reading-list entry](https://latent-implicit-thinking.github.io/papers/2610.35101/)
[Preprint BibTeX](https://latent-implicit-thinking.github.io/papers/2610.35101/citation.bib) · [Preprint CSL-JSON](https://latent-implicit-thinking.github.io/papers/2610.35101/citation.csl.json) · [JSON record](https://latent-implicit-thinking.github.io/papers/2610.35101/record.json)

## Approaches

- [Continuous-state reasoning](https://latent-implicit-thinking.github.io/topics/continuous-states/): Use vectors—lists of numbers—to carry intermediate reasoning from one step to the next, instead of selecting a text token at every step. Some methods use the model's hidden states; others combine several candidate token representations.
- [Alternative generation methods](https://latent-implicit-thinking.github.io/topics/alternative-reasoning-schedules/): Change how intermediate steps are generated, revised, or searched—for example, by refining several steps in parallel. Some produce written chains of thought.
- [Analysis and evaluation](https://latent-implicit-thinking.github.io/topics/implicit-reasoning-evaluation/): Measure how well these methods work, investigate what their internal computation does, and identify where they fail. This includes experiments, studies of model internals, and mathematical results.

Reading-list version 0.12.0; literature checked through 2026-11-02.