Does the signal reconstruction error in compressed sensing using $l_1$ norm minimization depends on the amplitude of non-zero coefficients and their location ?
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$\begingroup$ No,but the number of non zero coefficients has effect on the quality $\endgroup$– AbhishekFeb 4, 2016 at 19:29
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2 Answers
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The short answer is no.
Typically in Compressive Sensing, the reconstruction error is defined in terms of mean squared error (MSE):
$$ MSE = \frac{|| \hat{x} - x ||_2^2}{||x||_2^2} $$
where $$ || y ||_2^2 = \sum_{i=0}^{n-1} y_i^2 $$
For a sparse signal this will simply be
$$||y_{sparse}||_2^2 = \sum_{j \in S_{sparse}} y_i^2 $$
where $S_{sparse}$ is the set of non-zero coefficients in the vector $y$. Note that this is independent of the indices $j$ - a permutation will give the same MSE.
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$\begingroup$ Thanks for your kind reply. I think this one is only for location. What about the amplitude of non-zero elements? $\endgroup$– J CianMar 5, 2016 at 17:32
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Yes and No. If you consider your signal, noise free, meaning that all of the zero elements are really zero, then your answer is No, however though if the input noise is taken into the account, then the higher amplitude of the non-zero elements results in better signal reconstruction (high input SNR gives high output SNR).
