Academy of Mathematics and Systems Science, CAS
Colloquia & Seminars
| | Speaker: | Prof.Shidong Li，San Francisco State University
| | Inviter: | | Title:
| Stability and the necessary and sufficient condition for the unique solution of the tail-minimization approach in compressed sensing with frames | Time & Venue:
| 2021.07.23 14:00-15:00 腾讯会议ID：854 310 031 | Abstract:
| Sparse frame representations and sparse signal/channel recovery have immediate applications in 5G wireless communication, and in array signal processing. We present in this talk results of the $\ell_1$tail-minimization approach applied to sparse frame representations in compressed sensing, where $y=Af$ is the measurement, $f=Dx$, $D$ is a dictionary/frame, and $x$ is the sparse frame expansion coefficients. The tail-$\ell_1$-synthesis approach is analyzed in detail. A tail dictionary null space property (tail-DNSP) is shown to be necessary and sufficient for the unique recovery of $f$. Stability results for solutions to sparse frame representation problems via tail-minimization are then derived using the tail-DNSP and variations. Stronger results for real $A$ and $D$ are also given. These analyses and extensive numerical tests show that the tail-minimization substantially out-performs the standard$\ell_1$-synthesis approach. We also show by examples that $A$ can satisfy the DNSP or tail-DNSP where $AD$ fails NSP. In such cases unique recovery is guaranteed at the signal level $f$ while failing at the coefficient level $x$. It is also demonstrated that $A$ satisfying tail-DNSP is weaker than A satisfying DNSP.
Also presented includes an equivalence between the traditional$\ell_1$-analysis approach and a synthesis problem with a new synthesized measurement matrix $\tilde A$. As a result, there is nowa necessary and sufficient $tilde A$-NSP condition for the unique solution of the $\ell_1$-analysis problem, which has never been seen in the literature to the best of our knowledge.
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