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Sparse approximate solutions to linear systems. (English) Zbl 0827.68054
Summary: The following problem is considered: given a matrix A in m×n , (m rows and n columns), a vector b in m , and ε>0, compute a vector x satisfying |Ax-b| 2 ε if such exists, such that x has the fewest number of non-zero entries over all such vectors. It is shown that the problem is NP-hard, but that the well-known greedy heuristic is good in that it computes a solution with at most 18Opt(ε/2)|𝐀 + | 2 2 ln(|b| 2 /ε) non-zero entries, where Opt(ε/2) is the optimum number of nonzero entries at error ε/2, 𝐀 is the matrix obtained by normalizing each column of A with respect to the L 2 norm, and 𝐀 + is its pseudo-inverse.

68Q25Analysis of algorithms and problem complexity