Sparse representation for ℓp−αℓq minimization and uniform condition for the recovery of approximately k-sparse signals with prior support information

Abstract

In the frame of ${\ell }_p-\alpha {\ell }_q$ metric, a novel lemma of sparse representation is developed. Uniform sufficient conditions for the stable recovery of approximately $k$-sparse signals with partial support information are derived in different noise settings via weighted ${\ell }_p-\alpha {\ell }_q$ minimization method, and moreover, the reconstruction error bounds are precisely characterized. Specifying different values of the parameters $p\in (0,1],q\in [p,+\infty ]$ and $\alpha \in [0,1]$ in the new results leads to some important special cases, including the optimal results in terms of ${\ell }_1$ convex minimization, ${\ell }_p$ nonconvex minimization, ${\ell }_1-{\ell }_2$ minimization, ${\ell }_1-\alpha {\ell }_q$ minimization and ${\ell }_p-\alpha {\ell }_1$ minimization in the literature. A series of numerical experiments demonstrate the advantage of the new method for sparse recovery.