  Interesting data often concentrate on low dimensional smooth manifolds inside
a high dimensional ambient space. Random projections are a simple, powerful
tool for dimensionality reduction of such data. Previous works have studied
bounds on how many projections are needed to accurately preserve the geometry
of these manifolds, given their intrinsic dimensionality, volume and curvature.
However, such works employ definitions of volume and curvature that are
inherently difficult to compute. Therefore such theory cannot be easily tested
against numerical simulations to understand the tightness of the proven bounds.
We instead study typical distortions arising in random projections of an
ensemble of smooth Gaussian random manifolds. We find explicitly computable,
approximate theoretical bounds on the number of projections required to
accurately preserve the geometry of these manifolds. Our bounds, while
approximate, can only be violated with a probability that is exponentially
small in the ambient dimension, and therefore they hold with high probability
in cases of practical interest. Moreover, unlike previous work, we test our
theoretical bounds against numerical experiments on the actual geometric
distortions that typically occur for random projections of random smooth
manifolds. We find our bounds are tighter than previous results by several
orders of magnitude.
