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Assume I have a sparse vector $x$ whose few values are non-zeros, that vector is multiplied with sparse unitary matrix $Y$ giving $z = Yx$. Are there any advantages of detecting the sparse vector $x$ based on $z$ the in that case?

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  • $\begingroup$ I don't understand the question. "detection of $x$" requires that you have something that you're not sure contains $x$ or doesn't contain $x$. What is that? (context is key!) $\endgroup$
    – Marcus Müller
    Commented Jun 5, 2020 at 18:04

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You are referring here to sparse signal sampling and Reconstruction. You do not require the sampling matrix to be sparse. Infact we require the sampling matrix to follow the restrictive isometry property. And sparse matrices do not follow this property. Random matrices do. So no point having a sparse sampling matrix

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