Orthogonal Matching Pursuit: A Brownian Motion Analysis

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
=4
k
log(
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
=2
k
log(
n
-
k
) , is in fact sufficient for asymptotic recovery. More generally, when the sparsity level satisfies
k
_min
≤
k
≤
k
_max
but is unknown,
m
=2
k
_max
log(
n
-
k
_min
) 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
=2
k
log(
n
-
k
) exactly matches the number of measurements required by the more complex lasso method for signal recovery with a similar SNR scaling.