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I recently introduced myself into the field of CS, but I do not understand why some people try to find deterministic measurement matrices? If I am correct, gaussian random matrices are very powerful with their properties. They are easy to create, their reconstruction power is very good (having a low RIP constant) and so on.

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As far as I know there are two reasons:

  • First of all for practical implementation, usage of random matrices is very hard, so people try to come up with matrices that are fixed, this is thought to be helpful for designing efficient hardwares.
  • Secondly, people try to find some deterministic matrices which their structure can be exploited for faster signal recovery. Using a structured matrix, might pave the way for efficient recovery. The goal is to achieving something like FFT for compressive recovery.
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  • $\begingroup$ Thank you very much, this was helpful! If FFT is equivalent to deterministic matrices, what is equivalent to random matrices ? $\endgroup$ – IdleRunner Jul 11 '17 at 7:41
  • $\begingroup$ @IdleRunner glad it helped! you know the conventional sampling theory of Shanon was first on the basis of random Gaussian sampling, so random sampling is equivalent to statistical observation of a population (here the sparse signal). $\endgroup$ – MimSaad Jul 11 '17 at 8:04
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The sampling matrix needs to be known to both during compression and during reconstruction. This typically means having to keep the entire matrix on both sides. This may be very prohibitive for sensor systems, which often have quite limited memory. A non-random matrix may be stored with a fewer number of parameters, and generated on the fly.

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