Piecewise Sparse Recovery in Unions of Bases

Abstract: Sparse recovery is widely applied in many fields, since many signals or vectors can be sparsely represented under some frames or dictionaries. Most of fast algorithms at present are based on solving $l^0$ or $l^1$ minimization problems and they are efficient in sparse recovery. However, compared with the practical results, the theoretical sufficient conditions on the sparsity of the signal for $l^0$ or $l^1$ minimization problems and algorithms are too strict. \par In many applications, there are signals with certain structures as piecewise sparsity. Piecewise sparsity means that the sparse signal $\mathbf{x}$ is a union of several sparse sub-signals, i.e., $\mathbf{x}=(\mathbf{x}_1^T,\ldots,\mathbf{x}_N^T)^T$, corresponding to the matrix $A$ which is composed of union of bases $A=[A_1,\ldots,A_N]$. In this paper, we consider the uniqueness and feasible conditions for piecewise sparse recovery. We introduce the mutual coherence for the sub-matrices $A_i\ (i=1,\ldots,N)$ to study the new upper bounds of $|\mathbf{x}|_0$ (number of nonzero entries of signal) recovered by $l^0$ or $l^1$ optimizations. The structured information of measurement matrix $A$ is used to improve the sufficient conditions for successful piecewise sparse recovery and also improve the reliability of $l_0$ and $l_1$ optimization models on recovering global sparse vectors.