Approximation of points on low-dimensional manifolds via random linear projections

Iwen, Mark A.; Maggioni, Mauro
June 2013
Information & Inference: A Journal of the IMA;Jun2013, Vol. 2 Issue 1, p1
Academic Journal
This paper considers the approximate reconstruction of points ∈ ℝD which are close to a given compact d-dimensional submanifold ℳ of ℝD using a small number of linear measurements of . In particular, it is shown that a number of measurements of , which is independent of the extrinsic dimension D suffices for a highly accurate reconstruction of . Furthermore, it is also proved that all vectors which are sufficiently close to ℳ can be reconstructed with uniform approximation guarantees when the number of linear measurements of 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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