2024/06/05 by Andrea Montanari, Montanari, Andrea, Kangjie Zhou +1
Earth and Planetary Sciences · Engineering · Mathematics · #68Q87 #93E20 (Primary) 60K35 (Secondary) #FOS: Computer and information sciences #FOS: Mathematics #Geophysics and Gravity Measurements #Machine Learning (cs.LG) #Morphological variations and asymmetry #Optical Polarization and Ellipsometry #Optimization and Control (math.OC) #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2406.02970
openalex publication_date 2024/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given d-dimensional standard Gaussian vectors \boldsymbolx1,…, \boldsymbolxn, we consider the set of all empirical distributions of its m-dimensional projections, for m a fixed constant. Diaconis and Freedman (1984) proved that, if n/d→ ∞, all such distributions converge to the standard Gaussian distribution. In contrast, we study the proportional asymptotics, whereby n,d→ ∞ with n/d→ α∈ (0, ∞). In this case, the projection of the data points along a typical random subspace is again Gaussian, but the set \mathscrFm,α of all probability distributions that are asymptotically feasible as m-dimensional projections contains non-Gaussian distributions corresponding to exceptional subspaces. Non-rigorous methods from statistical physics yield an indirect characterization of \mathscrFm,α in terms of a generalized Parisi formula. Motivated by the goal of putting this formula on a rigorous basis, and to understand whether these projections can be found efficiently, we study the subset \mathscrF\rm algm,α⊆ \mathscrFm,α of distributions that can be realized by a class of iterative algorithms. We prove that this set is characterized by a certain stochastic optimal control problem, and obtain a dual characterization of this problem in terms of a variational principle that extends Parisi's formula. As a byproduct, we obtain computationally achievable values for a class of random optimization problems including `generalized spherical perceptron' models.