compressed sensing

• This is related to the concept of pseudo-inverse
• Say we have an under-determined linear equation: \(y=Ax\) where the dimension of \(x\) is greater than the dim of \(y\)
• There are an infinite number of solutions \(y\), but if we add an additional constraint such as sparsity, that is we require the solution to minimize the l2 norm, then there is a unique solution (this is the pseudo-inverse).
• What does this have to do with "compression" and "sensing". The "compression" part is because we can recover \(x\) from \(y\), if the space of all signals \(x\) is restricted to simply the "sparse" signals. The "sensing" part I am not sure.