A Sharp Restricted Isometry Constant Bound of Orthogonal Matching Pursuit
Abstract
We shall show that if the restricted isometry constant (RIC) $\delta_{s+1}(A)$ of the measurement matrix $A$ satisfies $$ \delta_{s+1}(A) < \frac{1}{\sqrt{s + 1}}, $$ then the greedy algorithm Orthogonal Matching Pursuit(OMP) will succeed. That is, OMP can recover every $s$-sparse signal $x$ in $s$ iterations from $b = Ax$. Moreover, we shall show the upper bound of RIC is sharp in the following sense. For any given $s \in \N$, we shall construct a matrix $A$ with the RIC $$ \delta_{s+1}(A) = \frac{1}{\sqrt{s + 1}} $$ such that OMP may not recover some $s$-sparse signal $x$ in $s$ iterations.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2015
- DOI:
- 10.48550/arXiv.1501.01708
- arXiv:
- arXiv:1501.01708
- Bibcode:
- 2015arXiv150101708M
- Keywords:
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- Computer Science - Information Theory;
- 15A23;
- 15A54;
- 42C40
- E-Print:
- 8 pages, submitted to the IEEE Transactions on Information Theory