Structured LISTA for Multidimensional Harmonic Retrieval

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.