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Performance analysis of tail-minimization and the linear rate of convergence of a proximal algorithm for sparse signal recovery

Meng Huang, Shidong Li

TL;DR

Recovery error bounds of tail-minimization and the rate of convergence of an efficient proximal alternating algorithm for sparse signal recovery are considered and the proposed algorithm significantly improves signal recovery performance compared to state-of-the-art techniques.

Abstract

Recovery error bounds of tail-minimization and the rate of convergence of an efficient proximal alternating algorithm for sparse signal recovery are considered in this article. Tail-minimization focuses on minimizing the energy in the complement $T^c$ of an estimated support $T$. Under the restricted isometry property (RIP) condition, we prove that tail-$\ell_1$ minimization can exactly recover sparse signals in the noiseless case for a given $T$. In the noisy case, two recovery results for the tail-$\ell_1$ minimization and the tail-lasso models are established. Error bounds are improved over existing results. Additionally, we show that the RIP condition becomes surprisingly relaxed, allowing the RIP constant to approach $1$ as the estimation $T$ closely approximates the true support $S$. Finally, an efficient proximal alternating minimization algorithm is introduced for solving the tail-lasso problem using Hadamard product parametrization. The linear rate of convergence is established using the Kurdyka-Łojasiewicz inequality. Numerical results demonstrate that the proposed algorithm significantly improves signal recovery performance compared to state-of-the-art techniques.

Performance analysis of tail-minimization and the linear rate of convergence of a proximal algorithm for sparse signal recovery

TL;DR

Recovery error bounds of tail-minimization and the rate of convergence of an efficient proximal alternating algorithm for sparse signal recovery are considered and the proposed algorithm significantly improves signal recovery performance compared to state-of-the-art techniques.

Abstract

Recovery error bounds of tail-minimization and the rate of convergence of an efficient proximal alternating algorithm for sparse signal recovery are considered in this article. Tail-minimization focuses on minimizing the energy in the complement of an estimated support . Under the restricted isometry property (RIP) condition, we prove that tail- minimization can exactly recover sparse signals in the noiseless case for a given . In the noisy case, two recovery results for the tail- minimization and the tail-lasso models are established. Error bounds are improved over existing results. Additionally, we show that the RIP condition becomes surprisingly relaxed, allowing the RIP constant to approach as the estimation closely approximates the true support . Finally, an efficient proximal alternating minimization algorithm is introduced for solving the tail-lasso problem using Hadamard product parametrization. The linear rate of convergence is established using the Kurdyka-Łojasiewicz inequality. Numerical results demonstrate that the proposed algorithm significantly improves signal recovery performance compared to state-of-the-art techniques.
Paper Structure (10 sections, 9 theorems, 121 equations, 5 figures, 1 table, 2 algorithms)

This paper contains 10 sections, 9 theorems, 121 equations, 5 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Performance Error Bounds
  3. Performance error bounds for constrained minimization
  4. The error bounds for tail-lasso
  5. Proximal HPP algorithm for solving tail-lasso problems
  6. The tail-HPP algorithm for tail-lasso
  7. A new proximal HPP algorithm for tail-Lasso
  8. Linear rate of convergence of PHPP
  9. Numerical experiments
  10. Discussions

Key Result

Lemma 2.2

For any disjoint subset $S, T \subset \left\{1,\ldots, n\right\}$ with $|S| \le s$ and $|T|\le k$, if the matrix $A\in {\mathbb R}^{m\times n}$ obeys order-$(s+k)$ restricted isometry property with constants $0<\delta_{s+k,l}<1$ and $\delta_{s+k,u}>0$, then holds for all ${\bm x}, {\bm x}' \in {\mathbb R}^n$ supported on $S$ and $T$, respectively. Here, $\delta_{s+k}=\max\left\{\delta_{s+k,l}, \d

Figures (5)

  • Figure 1: The diagram of sets $S, T$, and $S_0$, where $x_j$ are arranged in non-increasing order.
  • Figure 2: Recovery performance of tail-$\ell_1$ minimization and the classical $\ell_1$ minimization under different RIP constants recorded in the same set experiments: (a) The curve of RIP constant $\delta_{k,l}$ vs sparsity $k$; (b) The success rate vs sparsity in noiseless case. Here, the measurement matrix $A\in {\mathbb R}^{64\times 256}$.
  • Figure 3: Success rates. $n=256, m=64$ and sparsity $k\in \left\{2,4,\cdots,34\right\}$.
  • Figure 4: Success rates. $n=256, k=12$ and $m=0.1n, 0.12n, \ldots, 0.3n$.
  • Figure 5: Success rates for noisy measurements. $n=256, m=64$ and sparsity $k\in \left\{2,4,\cdots,34\right\}$.

Theorems & Definitions (30)

  • Definition 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • proof : Proof of Theorem \ref{['th:1']}
  • Theorem 2.6
  • Remark 2.7
  • Remark 2.8
  • proof : Proof of Theorem \ref{['th:0']}
  • ...and 20 more