I have a sparse vector $x$ of size $N$x$1$ which is sparse with number of non-zeros values are $m$, it means $m$ out of $N$ values are non-zeros, the non-zeros locations are distributed randomly.

Are we able to compress the vector $x$ to be of length $m$x$1$ using a matrix $A$ of size $N$x$m$ ? Generally, I think it's possible since we have the dictionary matrix $A$, But what should be $A$ ? I mean, how can we generate it? Assuming we have the vectors $x$ is of length $8$x$1$ with only two non-zeros values.

  • $\begingroup$ Is it possible to get a handle on the problem overall? If $x$ is sparse, then you can simply throw away the zeros (in fact, this is how very large sparse matrices are represented anyway). If $x$ is sparse in some domain, then it is worth designing the matrix $A$ that provides the data to reconstruct $x$. $\endgroup$ – A_A Jun 10 '20 at 10:06
  • $\begingroup$ If we acn simply throw away the zeros, so how can I know what is the original sparse vector? .. Second, how can I design the matrix $A$ ? $\endgroup$ – Gze Jun 10 '20 at 10:47

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