Friday, April 20, 2012

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

Here is something that appeared on the Arxiv radar screen: Approximation of Points on Low-Dimensional Manifolds Via Random Linear Projections by Mark A. Iwen, Mauro Maggioni. The abstract reads:
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.
The  Geometric Multi-Resolution Analysis Code is here.  Mauro  tells me that "...We will be uploading the code in the next few days, as part of an update to the Geometric Multi-Resolution code...". Stay tuned!

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