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Testing the nullspace property using semidefinite programming. (English) Zbl 1211.90167
Summary: Recent results in compressed sensing show that, under certain conditions, the sparsest solution to an underdetermined set of linear equations can be recovered by solving a linear program. These results either rely on computing sparse eigenvalues of the design matrix or on properties of its nullspace. So far, no tractable algorithm is known to test these conditions and most current results rely on asymptotic properties of random matrices. Given a matrix A, we use semidefinite relaxation techniques to test the nullspace property on A and show on some numerical examples that these relaxation bounds can prove perfect recovery of sparse solutions with relatively high cardinality.
MSC:
90C22Semidefinite programming
94A12Signal theory (characterization, reconstruction, filtering, etc.)
90C27Combinatorial optimization
Software:
SDPT3; DSPCA; SeDuMi
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