High Quality Content by WIKIPEDIA articles! Sparse approximation is the problem of finding a signal or vector estimate with sparseness property, that is having a small number of nonzero elements, that satisfies (approximately) a system of equations. For example, consider a linear system of equations y = Ax, where A is a real M-by-N matrix and M < N. In general, this problem is ill-posed as there are infinitely many x that solve this system. One way to enforce sparsity is to choose x such that as many components as possible are zero. In other words, we want to solve min_x |x|_0, text{ such that } y = A x, where the objective function is defined by |x|_0 = #{ k : x_k neq 0, , k=1,ldots,N } and # denotes the cardinality of the set. However, this problem is NP-complete because classical combinatorial optimization can be reduced to it.

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