We introduce a learning-based algorithm to obtain a measurement matrix for compressive sensing related recovery problems. The focus lies on matrices with a constant modulus constraint which typically represent a network of analog phase shifters in hybrid precoding/combining architectures. We interpret a matrix with restricted isometry property as a mapping of points from a high- to a low-dimensional hypersphere. We argue that points on the low-dimensional hypersphere, namely, in the range of the matrix, should be uniformly distributed to increase robustness against measurement noise. This notion is formalized in an optimization problem which uses one of the maximum mean discrepancy metrics in the objective function. Recent success of such metrics in neural network related topics motivate a solution of the problem based on machine learning. Numerical experiments show better performance than random measurement matrices that are generally employed in compressive sensing contexts. Further, we adapt a method from the literature to the constant modulus constraint. This method can also compete with random matrices and it is shown to harmonize well with the proposed learning-based approach if it is used as an initialization. Lastly, we describe how other structural matrix constraints, e.g., a Toeplitz constraint, can be taken into account, too.
翻译:我们引入了一种基于学习的算法,以获得压缩感应相关回收问题的测量矩阵。 重点是具有恒定模量限制的矩阵, 通常代表混合预编码/组合结构中模拟相位变器的网络。 我们将一个具有限制性等离子属性的矩阵解释为从高到低维超球点的绘图。 我们争辩说, 低维超球点, 即从矩阵范围, 应该统一分布, 以提高测量噪音的稳健性能。 这一概念被正式确定为优化问题, 优化问题使用目标函数中的最大平均值差异度指标之一。 神经网络相关专题最近的成功促使以机器学习为基础解决问题。 数值实验显示的性能优于在压缩感测环境中通常使用的随机测量矩阵。 此外, 我们从文献中将一种方法调整到恒定的模量限制。 这种方法还可以与随机矩阵竞争, 并且如果在初始使用以学习为基础的方法时, 显示该方法与拟议的基于学习的方法相协调。 最后, 我们描述了其他结构矩阵的限制, 如何被应用到结构矩阵的制约 。