用于多维谐波恢复的结构化LISTA
Structured LISTA for Multidimensional Harmonic Retrieval
摘要：
学习迭代收缩阈值算法（LISTA）采用深度学习技术从标记的训练数据中学习最佳算法参数，可成功应用于小规模多维谐波恢复（MHR）问题。然而，LISTA计算要求大规模的MHR问题，因为学习的互抑制矩阵的矩阵大小与信号长度呈二次增长。这些大型矩阵消耗昂贵的内存/计算资源，并需要大量的标记数据进行训练，限制了LISTA方法的适用性。在本文中，我们证明了MHR问题的互抑制矩阵自然具有Toeplitz结构，这意味着矩阵的自由度（DoF）可以从二次阶减少到线性阶。利用这一特性，我们提出了一种结构化的LISTA-Toeplitz网络，它对互抑制矩阵施加了Toeplitz结构限制，并采用线性卷积代替传统LISTA网络中涉及的矩阵向量乘法。通过仿真和雷达空中目标检测的外场试验，验证了该网络的性能。对于小规模的MHR问题，LISTAToeplitz表现出接近甚至更好的恢复精度比传统的LISTA，而前者显着降低了网络的复杂性，并需要更少的训练数据。对于大规模的MHR问题，LISTA是难以实现的，由于巨大的尺寸的相互抑制矩阵，我们提出的LISTA-Toeplitz仍然享有理想的恢复性能。
英文摘要：
Learned iterative shrinkage thresholding algorithm (LISTA), which adopts deep learning techniques to learn optimal algorithm parameters from labeled training data, can be successfully applied to small-scale multidimensional harmonic retrieval (MHR) problems. However, LISTA computationally demanding for large-scale MHR problems because the matrix size of the learned mutual inhibition matrix exhibits quadratic growth with the signal length. These large matrices consume costly memory/computation resources and require a huge amount of labeled data for training, restricting the applicability of the LISTA method. In this paper, we show that the mutual inhibition matrix of a MHR problem naturally has a Toeplitz structure, which means that the degrees of freedom (DoF) of the matrix can be reduced from a quadratic order to a linear order. By exploiting this characteristic, we propose a structured LISTA-Toeplitz network, which imposes a Toeplitz structure restriction on the mutual inhibition matrices and applies linear convolution instead of the matrix-vector multiplication involved in the traditional LISTA network. Both simulation and field test for air target detection with radar are carried out to validate the performance of the proposed network. For small-scale MHR problems, LISTAToeplitz exhibits close or even better recovery accuracy than traditional LISTA, while the former significantly reduces the network complexity and requires much less training data. For large-scale MHR problems, where LISTA is difficult to implement due to the huge size of the mutual inhibition matrices, our proposed LISTA-Toeplitz still enjoys desirable recovery performance.
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