Overparametrized linear dimensionality reductions: From projection pursuit to two-layer neural networks

TL;DR

This paper investigates the behavior of low-dimensional projections of high-dimensional Gaussian data in the asymptotic limit, providing bounds on the set of achievable distributions and applying these results to neural network interpolation thresholds.

Contribution

It introduces new bounds on the set of distributions from high-dimensional projections, characterizes the Wasserstein radius, and applies findings to neural network analysis.

Findings

Characterized the Wasserstein radius of projection distributions.

Established bounds for the set of achievable distributions in high dimensions.

Provided an upper bound on the interpolation threshold for two-layer neural networks.

Abstract

Given a cloud of $n$ data points in $\mathbb{R}^d$, consider all projections onto $m$-dimensional subspaces of $\mathbb{R}^d$ and, for each such projection, the empirical distribution of the projected points. What does this collection of probability distributions look like when $n,d$ grow large? We consider this question under the null model in which the points are i.i.d. standard Gaussian vectors, focusing on the asymptotic regime in which $n,d\to\infty$, with $n/d\to\alpha\in (0,\infty)$, while $m$ is fixed. Denoting by $\mathscr{F}_{m, \alpha}$ the set of probability distributions in $\mathbb{R}^m$ that arise as low-dimensional projections in this limit, we establish new inner and outer bounds on $\mathscr{F}_{m, \alpha}$. In particular, we characterize the Wasserstein radius of $\mathscr{F}_{m,\alpha}$ up to constant multiplicative factors, and determine it exactly for $m=1$. We also prove sharp bounds in terms of Kullback-Leibler divergence and R\'{e}nyi information dimension. The previous question has application to unsupervised learning methods, such as projection pursuit and independent component analysis. We introduce a version of the same problem that is relevant for supervised learning, and prove a sharp Wasserstein radius bound. As an application, we establish an upper bound on the interpolation threshold of two-layers neural networks with $m$ hidden neurons.