Approximation of Points on Low-Dimensional Manifolds Via Random Linear Projections

TL;DR

This paper demonstrates that points near a low-dimensional manifold in high-dimensional space can be accurately reconstructed using a small, dimension-independent number of linear measurements, with practical algorithms provided.

Contribution

It introduces a method for manifold-based signal recovery using random linear projections with guarantees independent of ambient dimension.

Findings

Number of measurements needed is independent of ambient dimension D.

Reconstruction accuracy is high with high probability.

A practical algorithm for manifold-based recovery is developed.

Abstract

This paper considers the approximate reconstruction of points, x \in R^D, which are close to a given compact d-dimensional submanifold, M, of R^D using a small number of linear measurements of x. In particular, it is shown that a number of measurements of x which is independent of the extrinsic dimension D suffices for highly accurate reconstruction of a given x with high probability. Furthermore, it is also proven that all vectors, x, which are sufficiently close to M can be reconstructed with uniform approximation guarantees when the number of linear measurements of x depends logarithmically on D. Finally, the proofs of these facts are constructive: A practical algorithm for manifold-based signal recovery is presented in the process of proving the two main results mentioned above.