Learning a Compressive Sensing Matrix with Structural Constraints via Maximum Mean Discrepancy Optimization
Learning a Compressive Sensing Matrix with Structural Constraints via Maximum Mean Discrepancy Optimization
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DOI：
10.1016/j.sigpro.2022.108553
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发表时间：
2021-10
期刊：
影响因子：
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通讯作者：
M. Koller;W. Utschick
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文献类型：
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作者：
M. Koller;W. Utschick
我们引入了一种基于学习的算法来获得与压缩感知相关的恢复问题的测量矩阵。重点在于具有恒模约束的矩阵，其例如表示混合预编码/组合体系结构中的模拟移相器网络。我们将具有受限等距性质的矩阵解释为从高维超球到低维超球的映射。我们认为低维超球面上的点应该均匀分布，以对抗测量噪声。这一概念被形式化为一个以最大平均差异度量为目标函数的优化问题。最近在神经网络相关主题中此类度量的成功促使了基于机器学习的优化问题的解决。数值实验表明，与压缩传感中典型的随机矩阵相比，该方法具有更好的性能。此外，我们将文献中的方法应用于常模数约束。该方法还可以与随机矩阵竞争，并且与所提出的算法具有很好的协调性。最后，我们描述了如何也考虑其他结构矩阵约束，例如Toeplitz约束。
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, e.g., represent a network of analog phase shifters in hybrid precoding/combining architectures. We interpret a matrix with restricted isometry property as a mapping from a high- to a low-dimensional hypersphere. We argue that points on the low-dimensional hypersphere should be uniformly distributed to combat measurement noise. This notion is formalized as an optimization problem which uses a maximum mean discrepancy metric as objective function. Recent success of such metrics in neural network related topics motivate a solution of the optimization problem based on machine learning. Numerical experiments show a better performance than random matrices that are typical for compressive sensing. Further, we adapt a method from the literature to the constant modulus constraint. This method can also compete with random matrices and harmonizes well with the proposed algorithm 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 as well.