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

Abstract: Given $d$-dimensional standard Gaussian vectors $\boldsymbol{x}<em>1,\dots, \boldsymbol{x}_n$, 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 $n/d\to \infty$, all such distributions converge to the standard Gaussian distribution. In contrast, we study the proportional asymptotics, whereby $n,d\to \infty$ with $n/d\to \alpha \in (0, \infty)$. In this case, the projection of the data points along a typical random subspace is again Gaussian, but the set $\mathscr{F}</em>{m,\alpha}$ 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 $\mathscr{F}<em>{m,\alpha}$ 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 $$\mathscr{F}^{\rm alg}</em>{m,\alpha}\subseteq \mathscr{F}_{m,\alpha}$$ 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.