We study the Compressed Sensing (CS) problem, which is the problem of finding the most sparse vector that satisfies a set of linear measurements up to some numerical tolerance. CS is a central problem in Statistics, Operations Research and Machine Learning which arises in applications such as signal processing, data compression and image reconstruction. We introduce an $\ell_2$ regularized formulation of CS which we reformulate as a mixed integer second order cone program. We derive a second order cone relaxation of this problem and show that under mild conditions on the regularization parameter, the resulting relaxation is equivalent to the well studied basis pursuit denoising problem. We present a semidefinite relaxation that strengthens the second order cone relaxation and develop a custom branch-and-bound algorithm that leverages our second order cone relaxation to solve instances of CS to certifiable optimality. Our numerical results show that our approach produces solutions that are on average $6.22\%$ more sparse than solutions returned by state of the art benchmark methods on synthetic data in minutes. On real world ECG data, for a given $\ell_2$ reconstruction error our approach produces solutions that are on average $9.95\%$ more sparse than benchmark methods, while for a given sparsity level our approach produces solutions that have on average $10.77\%$ lower reconstruction error than benchmark methods in minutes.

翻译：我们研究了压缩感知问题，即在一定数值容差下寻找满足线性测量方程组的最稀疏向量的优化问题。压缩感知是统计学、运筹学和机器学习领域的基础问题，广泛应用于信号处理、数据压缩和图像重建等场景。我们引入$\ell_2$正则化形式的压缩感知模型，并将其重构为混合整数二阶锥规划问题。通过推导该问题的二阶锥松弛，我们证明在正则化参数的温和约束条件下，所得松弛等价于经典的基础追踪去噪问题。进一步提出一种强化二阶锥松弛的半定松弛方法，并开发基于二阶锥松弛的定制分支定界算法，以实现压缩感知实例的可证明最优求解。数值实验表明，在合成数据上，我们的方法在数分钟内得到的解平均比当前最优基准方法稀疏6.22%；在真实心电图数据上，对于给定的$\ell_2$重构误差，本方法所得解比基准方法平均稀疏9.95%，而针对给定稀疏度水平，本方法所得解的重构误差比基准方法平均降低10.77%。