The possible numbers of nonzero entries in a matrix with a given term rank are determined respectively in the generic case, the symmetric case and the symmetric case with 0’s on the main diagonal. The matrices that attain the largest number of nonzero entries are also determined.

zero and nonzero entries in its local solutions is a very important task. However, most algorithms for solving the problem can only provide an approximate local optimal solution, where nonzero entries in the solution cannot be identified theoretically. In this paper, we establish lower bounds

This technical report demonstrates theoretically and empirically that a greedy algorithm called Orthogonal Matching Pursuit (OMP) can reliably recover a signal with m nonzero entries in dimension d given O(m ln d) random linear measurements of that signal. This is a massive improvement over

Abstract not found

A full-rank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity

) · S · log n, where S is the number of nonzero components in x 0, and µ is the largest entry in U properly normalized: µ(U) = √ n · maxk,j |Uk,j|. The smaller µ, the fewer samples needed. The result holds for “most ” sparse signals x 0 supported on a fixed (but arbitrary) set T. Given T, if the sign of x 0

only requires a matrix-vector product. The sparsity pattern of the approximate inverse is not imposed a priori but captured automatically. This keeps the amount of work and the number of nonzero entries in the preconditioner to a minimum. Rigorous bounds on the clustering of the eigenvalues

This article demonstrates theoretically and empirically that a greedy algorithm called Orthogonal Matching Pursuit (OMP) can reliably recover a signal with m nonzero entries in dimension d given O(m ln d) random linear measurements of that signal. This is a massive improvement over previous

to full or sparse matrices, without any explicit action by the user. The sparse data structure represents a matrix in space proportional to the number of nonzero entries, and most of the operations compute sparse results in time proportionaltothenumber of arithmetic operations on nonzeros.

We consider efficient methods for the recovery of block-sparse signals — i.e., sparse signals that have nonzero entries occurring in clusters—from an underdetermined system of linear equations. An uncertainty relation for block-sparse signals is derived, based on a block-coherence measure, which