If my measurement matrix have same number of row and column and the unknown vector is sparse can I still use Compressive Sensing to get better reconstruction with fewer measurement?

  • $\begingroup$ Fewer measurements compared to what? The number of measurements corresponds to the number of rows in the measurement matrix, which you claim is square. You can try to use sparse recovery methods to get a better estimate of your unknown sparse vector compared to trying to just solve the system of equations in the classical sense. $\endgroup$ – Florian Feb 6 '20 at 13:53
  • $\begingroup$ For example i can get a good estimation with a non sparse regularization method when the number of measurement (as you claim also number of rows in measurement matrix) is N. But cant get same good result when M<N measurement is used. So the question is if the unkown vector is sparse can i get better result with M measurements. As i saw in compressive sensing literature measurement matrix is not square which leads to underdetermined system of equations. Thats why i am asking if it has to be underdetermined system or square matrix is also possible. And maybe some explanation behind it. $\endgroup$ – johanson Feb 6 '20 at 14:45

It's important to clearly distinguish the terms Compressive Sensing (CS) and Sparse Signal Recovery (SSR). CS is about taking fewer measurements than what classical criteria such as Nyquist would dictate us. In the discrete setting this means taking $M<N$ measurements of an $N$-sparse vector.

SSR is about how to recover our $N$-sparse vector from your measurements.

That said, this should answer your question: If you apply CS, then you need SSR to recover your signal, however, the converse is not true - you can apply SSR even if you didn't do any subsampling. For SSR, your measurement matrix can be flat or square, it could even be tall.

Mathematically speaking, for SSR you solve a problem of the sort $$\min_x \|y - A x \| + \lambda h(x),$$ where $h(x)$ is a regularizer that encourages sparse solutions, i.e., an $\ell_0$-"norm" or its relaxation, the $\ell_1$-norm. In CS, such problems are solved since $A$ is flat and hence we must regularize to find our sparse solution out of the infinitely many possible solutions. The same problem can be used in a non-CS setup where $A$ is square or even tall, e..g, to find an approximate solution that shows a desired level of sparsity.

  • $\begingroup$ This is really hepful to me! Thanks. $\endgroup$ – johanson Feb 6 '20 at 19:28
  • $\begingroup$ what about the criterias should sensing matrix meet like restricted isometry hypothesis. Is it still important in SSR? $\endgroup$ – johanson Feb 6 '20 at 19:53
  • $\begingroup$ You'd want your matrix to be well-conditioned. RIP is something concerning flat matrices since square or tall matrices can have ideal RICs equal to zero. But conditioning of the matrix $A$ is of course still a relevant issue. $\endgroup$ – Florian Feb 7 '20 at 17:10
  • $\begingroup$ @Florian Could you send me any reference documents about SSR and matlab code example if possible. Because I tried to work with compressive sensing with square matrix but I get always residual error. I put my question here dsp.stackexchange.com/questions/64724/… . . So let's maybe try SSR. $\endgroup$ – Gze Mar 21 '20 at 14:26

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