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Exact sparse reconstruction form Vandermonde matrices

As a conclusion in classical linear algebra, an underdetermined linear equations usually have an infinite number of solutions. The sparest one among these solutions is significant in many applications. This problem can be modeled as the $l_0$-minimization, However, to find the sparsest solution of an underdetermined linear equations is NP-hard. Therefore, an important approach to solve the following $l_p$-minimization ($0<p\leq1$), The purpose of this problem is to find a $p$-norm minimization solution $(0<p\leq1)$ instead of the sparest one. In order to study the equivalence relationship between $l_0$-minimization and $l_p$-minimization, most of related work adopt Restricted Isometry Property (RIP) and Restricted Isometry Constant (RIC). On the premise of RIP and RIC, those work only solve the situation when the solution $\breve{x}$ of $l_0$-minimization satisfies that $\|\breve{x}\|_0<k$ where $k$ is a known fixed constant with $k<\frac{spark(A)}{2}$. One of the results in this paper is to give an analytic expression $p^*$ such that $l_p$-minimization is equivalent to $l_0$-minimization for every $\|\breve{x}\|_0<\frac{spark(A)}{2}$. In this paper, we also consider the case where the matrix $A$ is a Vandermonde matrix and we present an analytic expression $p^*$ such that the solution of $l_p$-minimization also solve $l_0$-minimization. Compared with the similar results based on RIP and RIC, we do not need the uniqueness assumption, i.e., the solution $x^*$ of $l_0$-minimization do not have to be assumed to be the unique solution which is the main breakthrough in our result. Another superiority of our result is its computability, i.e., each part in the analytic expression can be easily calculated.