Terahertz sparse deconvolution based on an iterative shrinkage and thresholding algorithm (ISTA) has been used to characterize multilayered structures with resolution equivalent to or finer than the sampling period of the measurement. However, this method was only studied on thin samples to separate the overlapped echos that can't be distinguished by other deconvolution algorithms. Besides, ISTA heavily depends on the convolution matrix consisting of delayed incident pulse, which is difficult to precisely extricate from the reference signal, and thereby fluctuations caused by noise are occasionally treated as echos. In this work, a terahertz sparse deconvolution based on a learned iterative shrinkage and thresholding algorithm (LISTA) is proposed. The method enclosed the matrix multiplication and soft thresholding in a block and cascaded multiple blocks together to form a deep network. The convolution matrices of the network were updated by stochastic gradient descent to minimize the distance between the output sparse vector and the optimal sparse representation of the signal, and subsequently the trained network made more precise estimation of the echos than ISTA. Additionally, LISTA is notably faster than ISTA, which is important for real-time tomographic-image processing. The algorithm was evaluated on terahertz tomographic imaging of a high-density poly ethylene (HDPE) sample, revealing obvious improvements in detecting defects of different sizes and depths. This technique has potential usage in nondestructive testings of thick samples, where echos reflected by minor defects are not discernible by existed deconvolution algorithms.