  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.
