The Geometry of Statistical Feature Learning in Mean-Field Langevin Dynamics

Abstract:We introduce a geometric formulation of statistical feature learning for supervised regression. Feature learning is defined through a base--fiber decomposition: the base is the feature-side geometry produced by training, and the fiber is the learned feature space where estimation is performed. We prove this property for spherical mean-field Langevin dynamics, viewed as the Wasserstein gradient flow of a negative entropy-regularized empirical risk. In Gaussian multi-index models, the low-temperature stationary distribution concentrates near the hidden indices, forms a multi-spike structure, and yields parameter recovery with high probability, even though negative entropy regularization penalizes concentration. This concentration has a sharp transition at temperature $\lambda\asymp 1$. In Gaussian single-index models, the stationary measure satisfies a concentration property, with parity determining whether it lives on $S_2^{d-1}$ or $\mathbb{RP}^{d-1}$. The induced learned feature space aligns the regression signal and yields rates $d/N$ and $Md/N$, up to logarithmic factors.

Statistics Theory (math.ST); Machine Learning (cs.LG); Machine Learning (stat.ML)

arXiv:2606.31429 [math.ST]

(or arXiv:2606.31429v2 [math.ST] for this version)

https://doi.org/10.48550/arXiv.2606.31429 Focus to learn more arXiv-issued DOI via DataCite

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