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Orthogonal Matching Pursuit: A Brownian Motion Analysis

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2 Author(s)
Fletcher, A.K. ; Dept. of Electr. Eng. & Comput. Sci., Univ. of California, Berkeley, CA, USA ; Rangan, S.

A well-known analysis of Tropp and Gilbert shows that orthogonal matching pursuit (OMP) can recover a k-sparse n-dimensional real vector from m=4klog(n) noise-free linear measurements obtained through a random Gaussian measurement matrix with a probability that approaches one as n→∞. This work strengthens this result by showing that a lower number of measurements, m=2klog(n-k) , is in fact sufficient for asymptotic recovery. More generally, when the sparsity level satisfies kminkkmax but is unknown, m=2kmaxlog(n-kmin) measurements is sufficient. Furthermore, this number of measurements is also sufficient for detection of the sparsity pattern (support) of the vector with measurement errors provided the signal-to-noise ratio (SNR) scales to infinity. The scaling m=2klog(n-k) exactly matches the number of measurements required by the more complex lasso method for signal recovery with a similar SNR scaling.

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Signal Processing, IEEE Transactions on  (Volume:60 ,  Issue: 3 )