This paper introduces a joint low-rank and sparsity-based model to address the problem of wall-clutter mitigation in compressed through-the-wall radar imaging. The proposed model is motivated by two observations that wall reflections reside in a low-rank subspace, and target signals tend to be sparse. In the proposed approach, the task of segregating target returns from wall reflections is formulated as a joint low-rank and sparsity constrained optimization problem. Here, the low rank constraint is imposed on the wall component and the sparsity constraint is used to model the target component. An iterative soft thresholding algorithm is developed to estimate a low-rank matrix of wall clutter and a sparse matrix of target reflections from a reduced measurement set. Once the wall and target components are estimated, the target signals are used for scene reconstruction. Experimental evaluation was conducted using real radar data. The results show that the proposed model is very effective at removing wall clutter and reconstructing the image of behind-the-wall targets from reduced measurements.