digital communications - Alternative to Orthogonal Matching Pursuit (OMP) Algorithm

In the Compressed Sensing context, assume there is a signal $ x \in {\mathbb{R}}^{n} $ which is $ k $ sparse. Namely its Pseudo $ {\ell}_{0} $ Norm is $ {\left\| x \right\|}_{0} = k $ (The signal has only $k $ non vanishing elements) where $ k << n $.

Given a Model Matrix $ A \in {\mathbb{R}}^{m \times n} $ the measurements are given by (This is the Model):

$$ y = A x $$

The recovery problem is given by:

$$ \arg \min_{x} {\left\| A x - y \right\|}_{2}^{2} \; \text{s. t.} \; {\left\| x \right\|}_{0} = k $$

Since the exact solution to the problem above is hard to find the recovery (Estimation) of the signal $ x $ from the measurements $ y $ is usually done using Orthogonal Matching Pursuit (OMP) Algorithm.

Basically the OMP finds iteratively the elements with highest correlation to the model.

The question is, since the measure of the quality to select indices is based on Correlation, why can't one just find the support using the following:

[vCorrVal vCorrIdx] = sort(A' * Y);    

vSignalSupport = vCorrIdx(1:k);

The result is almost the same as that of normal OMP.