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Difference-of-Convex Elastic Net for Compressed Sensing

Lang Yu, Nanjing Huang

Abstract

This work proposes a novel and unified sparse recovery framework, termed the difference of convex Elastic Net (DCEN). This framework effectively balances strong sparsity promotion with solution stability, and is particularly suitable for high-dimensional variable selection involving highly correlated features. Built upon a difference-of-convex (DC) structure, DCEN employs two continuously tunable parameters to unify classical and state-of-the-art models--including LASSO, Elastic Net, Ridge, and $\ell_1-α\ell_2$--as special cases. Theoretically, sufficient conditions for exact and stable recovery are established under the restricted isometry property (RIP), an oracle inequality and recovery bound are derived for the global solution, and a closed-form expression of the DCEN regularization proximal operator is obtained. Moreover, two efficient optimization algorithms are developed based on the DC algorithm (DCA) and the alternating direction method of multipliers (ADMM). Within the Kurdyka-Łojasiewicz (KŁ) framework, the global convergence of DCA and its linear convergence rate are rigorously established. Furthermore, DCEN is extended to image reconstruction by incorporating total variation (TV) regularization, yielding the DCEN-TV model, which is efficiently solved via the Split Bregman method. Numerical experiments demonstrate that DCEN consistently outperforms state-of-the-art methods in sparse signal recovery, high-dimensional variable selection under strong collinearity, and Magnetic Resonance Imaging (MRI) image reconstruction, achieving superior recovery accuracy and robustness.

Difference-of-Convex Elastic Net for Compressed Sensing

Abstract

This work proposes a novel and unified sparse recovery framework, termed the difference of convex Elastic Net (DCEN). This framework effectively balances strong sparsity promotion with solution stability, and is particularly suitable for high-dimensional variable selection involving highly correlated features. Built upon a difference-of-convex (DC) structure, DCEN employs two continuously tunable parameters to unify classical and state-of-the-art models--including LASSO, Elastic Net, Ridge, and --as special cases. Theoretically, sufficient conditions for exact and stable recovery are established under the restricted isometry property (RIP), an oracle inequality and recovery bound are derived for the global solution, and a closed-form expression of the DCEN regularization proximal operator is obtained. Moreover, two efficient optimization algorithms are developed based on the DC algorithm (DCA) and the alternating direction method of multipliers (ADMM). Within the Kurdyka-Łojasiewicz (KŁ) framework, the global convergence of DCA and its linear convergence rate are rigorously established. Furthermore, DCEN is extended to image reconstruction by incorporating total variation (TV) regularization, yielding the DCEN-TV model, which is efficiently solved via the Split Bregman method. Numerical experiments demonstrate that DCEN consistently outperforms state-of-the-art methods in sparse signal recovery, high-dimensional variable selection under strong collinearity, and Magnetic Resonance Imaging (MRI) image reconstruction, achieving superior recovery accuracy and robustness.
Paper Structure (18 sections, 17 theorems, 69 equations, 4 figures, 3 tables, 4 algorithms)

This paper contains 18 sections, 17 theorems, 69 equations, 4 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Recovery properties of CDCEN
  3. Exact and stable recovery
  4. Consistency Theory for DCEN
  5. Proximal Operator
  6. DC algorithm
  7. DC decomposition and iteration framework
  8. ADMM solver for the subproblem in DCEN-DCA
  9. Convergence analysis
  10. Global convergence and convergence rate
  11. ADMM for Solving DCEN
  12. Numerical experiments
  13. Parameter settings
  14. Efficiency Evaluation in Noiseless Scenario
  15. Robust recovery of DCEN
  16. ...and 3 more sections

Key Result

Lemma 1

Let $\alpha \in (0,1)$ and $\bm{x} \in \mathbb{R}^n \setminus \{\bm{0}\}$ with $\|\bm{x}\|_0 = s$ and $\Lambda = \mathrm{supp}(\bm{x})$. Then the following statements hold: where $x_{\min}$ in (a) is defined as $x_{\min} = \min_j |x_j|$ and $x_{\min}$ in (b) is defined as $x_{\min} = \min_{j \in \Lambda} |x_j|$. $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure 1: Relative error distributions (log10) of LASSO, $\ell_1-\alpha\ell_2$, and DCEN models under DCT/Gaussian sensing matrices with varying $F$, $r$, and $s$.
  • Figure 2: Success rate versus sparsity $s$ for oversampled DCT.
  • Figure 3: MRI reconstruction results from 8 radial sampled projections.
  • Figure 4: Variable selection stability under highly correlated predictors ($n=20$, $p=100$; averaged over 100 Monte Carlo trials).

Theorems & Definitions (38)

  • Lemma 1
  • proof
  • Remark 2
  • Lemma 3
  • proof
  • Remark 4
  • Lemma 5
  • proof
  • Theorem 6
  • proof
  • ...and 28 more