arXiv：cs.LG· Swagatam Das·· 4 小时前AI 评分35
超越半圆律：具有指定平衡态的自由扩散模型
Beyond the Semicircle: Free Diffusion Models with Prescribed Equilibria
AI 导读
研究提出用状态依赖的自由波动率替代常系数自由扩散，通过闭式 Hilbert 变换漂移为任意足够规则的紧支撑目标分布构造以其为平稳谱分布的自由 Fokker–Planck 流。在凸势条件下该构造具备全局松弛的扩散解释、反向时间动力学与可训练 score，且无法通过变量变换化归为常系数情形。作者在 Marchenko–Pastur 律上实例化并数值验证，实现了维度无关的精确 score 与端到端生成。
正文
Abstract:A growing class of machine-learning objects -- covariance and Gram matrices, kernel and attention matrices, MIMO channel matrices, density operators -- are naturally spectra rather than coordinate vectors. Building a denoising diffusion model for such data by corrupting eigenvalues coordinatewise is not merely elegant: it converges to the wrong limit, because eigenvalues repel rather than move independently. Free probability theory supplies the correct forward process, with free convolution replacing classical convolution and Voiculescu's conjugate variable replacing the score, but constant-coefficient free diffusions have a hidden limitation of their own: their only possible equilibrium is the semicircular law, whatever the target distribution looks like. We show that state-dependent free volatility removes this restriction. For any sufficiently regular compactly supported target law, we explicitly construct, through a closed-form Hilbert-transform drift, a free Fokker--Planck flow having that law as a stationary spectral distribution. Under an additional convex-potential condition, the construction admits a globally relaxing diffusion interpretation, reverse-time dynamics, and a trainable score. The generalization is genuine rather than cosmetic: no change of spectral variable reduces it to the constant-coefficient case, and it is a Wasserstein gradient flow only for an explicitly characterized family of coefficients that contains no bounded non-constant member. We instantiate the construction on the Marchenko--Pastur law, the limiting spectrum of isotropic sample covariance matrices, and validate it numerically: the free-versus-coordinatewise gap, an exact dimension-independent score driving reverse-time matrix dynamics at several matrix sizes, the designed Marchenko--Pastur equilibrium, and end-to-end generation with a learned score.
| Subjects: | Probability (math.PR); Machine Learning (cs.LG); Machine Learning (stat.ML) |
| Cite as: | arXiv:2510.22778 [math.PR] |
| (or arXiv:2510.22778v4 [math.PR] for this version) | |
| https://doi.org/10.48550/arXiv.2510.22778 arXiv-issued DOI via DataCite |
Submission history
From: Swagatam Das [view email]
[v1]
Sun, 26 Oct 2025 18:03:54 UTC (83 KB)
[v2]
Sun, 2 Nov 2025 19:17:12 UTC (85 KB)
[v3]
Thu, 6 Aug 2026 19:04:35 UTC (1,238 KB)
[v4]
Mon, 5 Oct 2026 18:54:09 UTC (1,563 KB)
来源：arXiv：cs.LG · arxiv.org