# Which exceptional low-dimensional projections of a Gaussian point cloud can be found in polynomial time?

arxiv（2024）

Abstract

Given d-dimensional standard Gaussian vectors x_1,…,
x_n, 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 ℱ_m,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 ℱ_m,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 ℱ^ alg_m,α⊆ℱ_m,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.

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