Random projections of random manifolds

TL;DR

This paper derives explicit, high-probability bounds on the number of random projections needed to preserve the geometry of Gaussian random manifolds, validated through numerical experiments and tighter than previous bounds.

Contribution

It provides the first explicit, computable bounds on projection counts for Gaussian manifolds, validated by experiments, improving upon prior theoretical results.

Findings

Bounds are exponentially small in ambient dimension.

Theoretical bounds are tighter than previous results by several orders of magnitude.

Numerical experiments confirm the high-probability validity of the bounds.

Abstract

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.