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arXiv · 1204.3337

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

Mark A. Iwen·Mauro Maggioni
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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.

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BibTeXRIS

Mark A. Iwen, Mauro Maggioni. 2012-04-16. Approximation of Points on Low-Dimensional Manifolds Via Random Linear Projections. https://arxiv.org/abs/1204.3337

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