This paper studies the task of estimating heterogeneous treatment effects in causal panel data models with covariate effects. We propose a novel Covariate-Adjusted Deep Causal Learning (CoDEAL) framework that cohesively deals with the underlying heterogeneity and nonlinearity of both panel units and covariate effects. CoDEAL integrates nonlinear covariate effect components (parameterized by a feed-forward neural network) with nonlinear factor structures (modeled by a multi-output autoencoder) to form a heterogeneous causal panel model. The nonlinear covariate component flexibly captures complex covariate influences on outcomes, and the nonlinear factor decomposition enables CoDEAL to effectively capture both cross-sectional and temporal dependencies inherent in the data panel. This latent structural information is subsequently integrated into a customized matrix completion algorithm, thereby facilitating more accurate counterfactual imputation. Moreover, the use of a multi-output autoencoder explicitly accounts for heterogeneity across units and enhances the model interpretability of the latent factors. We establish theoretical guarantees on the convergence of the estimated counterfactuals and demonstrate the compelling performance of the proposed method using extensive simulation studies and real-world data applications.
Conventional matrix completion methods exploit the underlying low-rank structure of a target matrix to recover its missing entries, but may be inadequate for capturing complex nonlinear dependencies. Deep neural networks have therefore been increasingly adopted for nonlinear matrix completion (NLMC). However, their high representational capacity, particularly in deep fully connected neural networks (FCNNs), can lead to over-fitting when training is restricted to incomplete observations. In this paper, we propose a Deep Neural Network with Nonsmooth Regularization (DNN-NSR) framework to address this issue by incorporating two complementary structural priors: sparsity of intermediate representations and low-rankness of layer-wise weight matrices. Specifically, the $\ell_1$-norm of hidden representations promotes sparse latent features, while the nuclear norm of weight matrices encourages low-rank layer-wise transformations. To optimize the resulting nonsmooth and nonconvex objective, we develop an extrapolated proximal gradient algorithm that employs soft-thresholding and singular-value thresholding to handle the two regularization terms, respectively. We further introduce a gradual regularization strategy that progressively increases the influence of these structural penalties during training. Under suitable assumptions, we establish convergence properties of the proposed algorithm toward a critical point of the training objective. The effectiveness of DNN-NSR is evaluated through extensive experiments on synthetic matrix recovery, single-image inpainting, and recommender systems using the MovieLens 100K and MovieLens 1M datasets. Comparisons with representative linear and nonlinear matrix completion methods demonstrate the potential of the proposed framework to improve recovery performance by jointly exploiting sparse latent representations and low-rank network transformations.