  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 logarithmic 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.
