Two Conditions for Equivalence of 0-Norm Solution and 1-Norm Solution in Sparse Representation

In sparse representation, two important sparse solutions, the 0-norm and 1-norm solutions, have been receiving much of attention. The 0-norm solution is the sparsest, however it is not easy to obtain. Although the 1-norm solution may not be the sparsest, it can be easily obtained by the linear programming method. In many cases, the 0-norm solution can be obtained through finding the 1-norm solution. Many discussions exist on the equivalence of the two sparse solutions. This paper analyzes two conditions for the equivalence of the two sparse solutions. The first condition is necessary and sufficient, however, difficult to verify. Although the second is necessary but is not sufficient, it is easy to verify. In this paper, we analyze the second condition within the stochastic framework and propose a variant. We then prove that the equivalence of the two sparse solutions holds with high probability under the variant of the second condition. Furthermore, in the limit case where the 0-norm solution is extremely sparse, the second condition is also a sufficient condition with probability 1.