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.