Wednesday, July 20, 2016

Random projections of random manifolds

Looks like some bounds have been tightened. Funny enough, in a matter of a week, I am either talking to or featuring someone from SpaceX, which just happens to deliver today or tomorrow a nanopore sequencer to the international space station

Small worlds!

Anyway, getting back to the paper.

Random projections of random manifolds by Subhaneil Lahiri, Peiran Gao, Surya Ganguli
A ubiquitous phenomenon is that interesting signals or data concentrate on low dimensional smooth manifolds inside a high dimensional ambient Euclidean space. Random projections are a simple and powerful tool for dimensionality reduction of such signals and data. Previous, seminal works have studied bounds on how the number of projections needed to preserve the geometry of these manifolds, at a given accuracy, scales with the intrinsic dimensionality, volume and curvature of the manifold. 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 quantitatively understand the tightness of the proven bounds. We instead study the typical distortions arising in random projections of an ensemble of smooth Gaussian random manifolds. In doing so, we find explicitly computable, approximate theoretical bounds on the number of projections required to preserve the geometric structure of these manifolds to a prescribed level of accuracy. 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 most 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. Through this comparison, we find our bounds are tighter than previous results by several orders of magnitude.

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