Transformed $\ell_p$ Minimization Model and Sparse Signal Recovery

TL;DR

The TLp penalty, which includes two aforementioned adjustable parameters, is introduced, offering more flexibility and stronger sparsity-promotion capability of the TLp minimization model, compared with the $\ell_p$ and the TL1 minimization models.

Abstract

In this article, we introduce a minimization model via a non-convex transformed $\ell_p$ (TLp) penalty function with two parameters $a\in(0,\infty)$ and $p\in(0,1]$, where the case $p=1$ is known and was established by S. Zhang and J. Xin. Using the sparse convex-combination technique, we establish the exact and the stable sparse signal recovery based on the restricted isometry property (RIP). We apply a modified iteratively re-weighted least squares method and the difference of convex functions algorithm (DCA) to give the IRLSTLp algorithm for unconstrained TLp minimization and prove some related convergence results. Finally, we conduct some numerical experiments to show the robustness of the IRLSTLp and the flexibility of the TLp minimization model. The novelty of these results lies in three aspects: (i) We introduce the concept of the relaxation degree RD$_P$ of a separable penalty function $P$ to quantitatively measure how closely $P$ approaches $\ell_0$. (ii) We introduce the TLp penalty, which includes two aforementioned adjustable parameters, offering more flexibility and stronger sparsity-promotion capability of the TLp minimization model, compared with the $\ell_p$ and the TL1 minimization models. (iii) The obtained RIP upper bound for signal recovery via TLp minimization can reduce, when $p\in(0,1]$ and as $a\to \infty$, to the sharp RIP bound obtained by R. Zhang and S. Li and, especially, can recover, when $p=1$, the well-known sharp bound $δ_{2s}<\frac{\sqrt{2}}{2}$.

Figures (9)

• Figure 1: Exact recovery for sparse $e$
• Figure 2: Level lines of $P_{0.1,p}$ with $p\in\{0.5,0.7,0.9,1\}$
• Figure 3: Level lines of $P_{a,0.7}$ with $a\in\{0.5,1,10\}$ and $P_{0.7}$
• Figure 4: Level lines of $P_{5,0.7}$, $\ell_5^{0.7}$, $P_{1,0.7}$, and $\ell_1^{0.7}$
• Figure 5: Numerical tests on $p$ and $a$ by $64\times 256$ Gaussian matrix

Theorems & Definitions (35)

• Definition 2.1
• Remark 2.2
• Example 2.3
• Lemma 2.4
• proof
• Proposition 2.5
• Definition 2.6
• Theorem 2.7
• Theorem 2.8
• Theorem 2.9