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In many applications, including climate science, power systems, and remote sensing, multiple input variables are observed for each output variable and the output variables are dependent. Several methods have been proposed to improve prediction by learning the conditional distribution of the output variables. However, when the relationship between the raw features and the outputs is nonlinear, the existing methods cannot capture both the nonlinearity and the underlying structure well. In this study, we propose a structured model containing hidden variables, which are nonlinear functions of inputs and which are linearly related with the output variables. The parameters modeling the relationships between the input and hidden variables, between the hidden and output variables, as well as among the output variables are learned simultaneously. To demonstrate the effectiveness of our proposed method, we conducted extensive experiments on eight synthetic datasets and three real-world challenging datasets: forecasting wind power, forecasting solar energy, and forecasting precipitation over U.S. The proposed method was more accurate than state-of-the-art structured

Joint Learning of Representation and Structure for Sparse Regression on Graphs