论文标题
基于L1/L2最小化的某些压缩感测模型的分析和算法
Analysis and algorithms for some compressed sensing models based on L1/L2 minimization
论文作者
论文摘要
最近，在一系列论文[32,38,39,41]中，提出了$ \ ell_1 $和$ \ ell_2 $ norms的比率作为无噪声压缩感应的稀疏功能。在本文中，我们进一步研究了这种模型在无噪声环境中的属性，并提出了一种算法，以最大程度地减少$ \ ell_1 $/$ \ ell_2 $在测量中受噪声的约束。具体而言，我们表明[32]中模型的扩展目标函数（约束集的目标和指标函数的总和）满足具有指数1/2的kurdyka-lojasiewicz（kl）属性；这使我们能够建立[39，等式中提出的算法的线性收敛。 11]在轻度假设下。接下来，我们扩展了$ \ ell_1 $/$ \ ell_2 $型号，以处理噪声的压缩感知问题。我们在球形部分属性[37,44]下为其中一些模型建立了解决方案存在，并扩展了[39，等式中的算法。 11]通过纳入移动助理 - 抗氧化技术[4]来解决这些问题。我们证明了在温和条件下我们的算法的随后收敛，并通过对特殊构建的潜在函数施加其他KL和可不同性假设来确定我们算法产生的整个序列的全局收敛。最后，我们通过通过我们的Algorithm求解相应的$ \ ell_1 $/$ \ ell_2 $模型来对稳健的压缩感应和基础追踪降级进行固定denoising denoising denoising denoising。我们的数值模拟表明，我们的算法能够以合理的精度恢复原始的稀疏向量。
Recently, in a series of papers [32,38,39,41], the ratio of $\ell_1$ and $\ell_2$ norms was proposed as a sparsity inducing function for noiseless compressed sensing. In this paper, we further study properties of such model in the noiseless setting, and propose an algorithm for minimizing $\ell_1$/$\ell_2$ subject to noise in the measurements. Specifically, we show that the extended objective function (the sum of the objective and the indicator function of the constraint set) of the model in [32] satisfies the Kurdyka-Lojasiewicz (KL) property with exponent 1/2; this allows us to establish linear convergence of the algorithm proposed in [39, Eq. 11] under mild assumptions. We next extend the $\ell_1$/$\ell_2$ model to handle compressed sensing problems with noise. We establish the solution existence for some of these models under the spherical section property [37,44], and extend the algorithm in [39, Eq. 11] by incorporating moving-balls-approximation techniques [4] for solving these problems. We prove the subsequential convergence of our algorithm under mild conditions, and establish global convergence of the whole sequence generated by our algorithm by imposing additional KL and differentiability assumptions on a specially constructed potential function. Finally, we perform numerical experiments on robust compressed sensing and basis pursuit denoising with residual error measured by $ \ell_2 $ norm or Lorentzian norm via solving the corresponding $\ell_1$/$\ell_2$ models by our algorithm. Our numerical simulations show that our algorithm is able to recover the original sparse vectors with reasonable accuracy.
