论文标题
非线性模型通过分段有理优化的稀疏信号重建
Sparse Signal Reconstruction for Nonlinear Models via Piecewise Rational Optimization
论文作者
论文摘要
我们提出了一种通过非线性失真降解并以有限的采样率获取的方法来重建稀疏信号。我们的方法将重建问题提出为数据拟合项和惩罚项的非概念最小化。与大多数以前为近似本地解决方案的工作相反，我们寻求一种全球解决方案，以解决获得的具有挑战性的非凸问题。我们的全球方法依赖于所谓的多项式优化的套索松弛。我们在这里特别包括分段有理功能的情况，这使得解决了$ \ ell_0 $惩罚功能的一系列非convex精确和连续放松。此外，我们研究了优化问题的复杂性。它显示了如何使用问题的结构有效减轻计算负担。最后，数值模拟说明了我们方法在全球最优性和信号重建方面的好处。
We propose a method to reconstruct sparse signals degraded by a nonlinear distortion and acquired at a limited sampling rate. Our method formulates the reconstruction problem as a nonconvex minimization of the sum of a data fitting term and a penalization term. In contrast with most previous works which settle for approximated local solutions, we seek for a global solution to the obtained challenging nonconvex problem. Our global approach relies on the so-called Lasserre relaxation of polynomial optimization. We here specifically include in our approach the case of piecewise rational functions, which makes it possible to address a wide class of nonconvex exact and continuous relaxations of the $\ell_0$ penalization function. Additionally, we study the complexity of the optimization problem. It is shown how to use the structure of the problem to lighten the computational burden efficiently. Finally, numerical simulations illustrate the benefits of our method in terms of both global optimality and signal reconstruction.
