The study of non-convex penalties has recently received considerable attentions in sparse approximation. The existing non-convex penalties are proposed on the principle of seeking for a continuous alternative to the ℓ₀-norm penalty. In this paper, we come up with a cubic spline penalty (CSP) which is also continuous but closer to ℓ₀-norm penalty compared to the existing ones. As a result, it produces the weakest bias among them. Wavelet tight frames are efficient for sparse approximation due to its redundancy and fast implementation algorithm. We adopt a tight frame balanced model with our proposed cubic spline penalty since the balanced model takes the advantages of both analysis and synthesis model. To solve the non-convex CSP penalized problem, we employ a proximal local linear approximation (PLLA) algorithm and prove the generated sequence converges to a stationary point of the model if it is bounded. Under additional conditions, we find that the limit point behaves as well as the oracle solution, which is obtained by using the exact support of the ground truth signal. The efficiency of our cubic spline penalty are further demonstrated in applications of variable selection and image deblurring.

中文翻译：

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紧框架平衡模型下稀疏近似的三次样条惩罚

非凸罚分的研究最近在稀疏近似中引起了相当大的关注。现有的非凸的处罚提出了寻求连续替代的原则ℓ ₀范数的惩罚。在本文中，我们提出了一个三次样条惩罚（CSP），这也是连续的，但更接近ℓ ₀-与现有标准相比的标准罚款。结果，它在其中产生了最弱的偏差。小波紧帧由于其冗余和快速实现算法而对于稀疏近似是有效的。由于平衡模型兼具分析模型和综合模型的优点，因此我们采用具有拟议三次样条罚分的紧框架平衡模型。为了解决非凸CSP的惩罚问题，我们采用了近端局部线性逼近（PLLA）算法，并证明了生成的序列在有界时会收敛到模型的固定点。在其他条件下，我们发现极限点的行为与预言解决方案一样好，这是通过使用地面真实信号的精确支持而获得的。