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The uniqueness of sparsest solutions of underdetermined linear systems playsa fundamental role in the newly developed compressed sensing theory. Severalnew algebraic concepts, including the sub-mutual coherence, scaled mutualcoherence, coherence rank, and sub-coherence rank, are introduced in this paperin order to develop new and improved sufficient conditions for the uniquenessof sparsest solutions. The coherence rank of a matrix with normalized columnsis the maximum number of absolute entries in a row of its Gram matrix that areequal to the mutual coherence. The main result of this paper claims that whenthe coherence rank of a matrix is low, the mutual-coherence-based uniquenessconditions for the sparsest solution of a linear system can be improved.Furthermore, we prove that the Babel-function-based uniqueness can be alsoimproved by the so-called sub-Babel function. Moreover, we show that thescaled-coherence-based uniqueness conditions can be developed, and that theright-hand-side vector $b$ of a linear system, the support overlap ofsolutions, the orthogonal matrix out of the singular value decomposition of amatrix, and the range property of a transposed matrix can be also integratedinto the criteria for the uniqueness of the sparsest solution of anunderdetermined linear system.

New and Improved Conditions for Uniqueness of Sparsest Solutions of Underdetermined Linear Systems