Structured LISTA for Multidimensional Harmonic Retrieval

Rong Fu, Yimin Liu, Tianyao Huang, Yonina C. Eldar

Abstract: Learned iterative shrinkage thresholding algorithm (LISTA), which adopts deep learning techniques to optimize algorithm parameters from labeled training data, can be successfully applied to small-scale multidimensional harmonic retrieval (MHR) problems. However, LISTA becomes computationally demanding for large-scale MHR 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. For MHR problems, the mutual inhibition matrix naturally has a Toeplitz structure, implying the degrees of freedom of the matrix can be reduced from quadratic order to linear order. We thereby propose a structured LISTA-Toeplitz network, which imposes Toeplitz structure on the mutual inhibition matrices and applies linear convolution instead of matrix-vector multiplications in traditional LISTA. Both simulation and field tests for air target detection with radar are carried out to validate the performance of the proposed network. For small-scale MHR problems, LISTA-Toeplitz 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 matrices, our proposed LISTA-Toeplitz still enjoys good recovery performance.