# Compressing sparse vectors based on compressive sensing

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.

• 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 $x$. – A_A Jun 10 '20 at 10:06
• 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 $A$ ? – Gze Jun 10 '20 at 10:47