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I have a sparse vector $x \in \mathbb{R}^{N \times 1}$, it's real and positive, the non-zeros values are maximum $N/2$ values. It means, I have at least $N/2$ zeros values in $x$.

My question, is it possible to build a measurement matrix $M$ with dimension $N/2 \times N$ as measurement matrix, where the dimension of the vector $x$ can be reduced into $N/2 \times 1$ real-positive values too?

How can I generate the matrix $M$ such that $M \times x = y$? The indices of non-zeros values in $x$ are not well-known; what is known that at least $N/2$ zeros values are existed in $x$. Is that feasible? what's the process to do that, I mean with which compressive sensing technique I can handle that problem?

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  • $\begingroup$ In the preamble you say $x \in \mathbb{R}^{N \times 1}$ and then later you say $x \in \mathbb{R}$. Do you really mean $x$ is as in the preamble and $y \in \mathbb{R}^{N/2 \times 1}$? Do you know something about the non-zero $x$ indices? If so, $\mathbf{M}$ is just a matrix with mostly zeros and ones where the measurement is extracted from $x$. $\endgroup$ Commented Jun 4, 2022 at 21:11
  • $\begingroup$ @PeterK. yes, I mean $x \in \mathbb{R}^{N \times 1}$ and $y \in \mathbb{R}^{N/2 \times 1}$. The non-zeros $x$ indices are changing randomly, their positions are not well-known but what I am sure of is that at least $N/2$-zeros are exist. (I update the questions following your notice too) $\endgroup$ Commented Jun 4, 2022 at 22:25

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For the specific realization of this specific $ \boldsymbol{x} $ you can easily chose the subset of columns matching the indices of the non zero elements of $ \boldsymbol{x} $.

Yet remember that the model matrix should handle multiple realizations of the signals. Hence it is usually a "fat" matrix.
So different signals can use different atoms (Columns).

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