Perturbations of Compressed Data Separation With Redundant Tight Frames

In the era of big data, the multi-modal data can be seen everywhere. Research on such data has attracted extensive attention in the past few years. In this paper, we investigate the perturbations of compressed data separation with redundant tight frames via <inline-formula> <tex-math notation="LaTeX">$\tilde {\boldsymbol {\Phi }}\text {-}{\ell _{q}}$ </tex-math></inline-formula>-minimization. By exploiting the properties of the redundant tight frame and the perturbation matrix, i.e., mutual coherence, null space property, and restricted isometry property, the condition on reconstruction of sparse signal with redundant tight frames is established, and the error estimation between the local optimal solution and the original signal is also provided. Numerical experiments are carried out to show that <inline-formula> <tex-math notation="LaTeX">$\tilde {\boldsymbol {\Phi }}\text {-}{\ell _{q}}$ </tex-math></inline-formula>-minimization is robust and stable for the reconstruction of sparse signal with redundant tight frames. To our knowledge, our works may be the first study concerning the perturbations of the measurement matrix and the redundant tight frame for compressed data separation.