Sparse multivariate regression with missing values in both predictors and responses

In statistical approaches to materials science, it is often necessary to predict multiple material properties simultaneously based on a limited number of experimental observations. However, due to the cost and difficulty in collecting comprehensive data, datasets often have small sample sizes and frequently contain missing values in both predictor and response variables. These issues are further compounded when the dimensionality of the variables is moderate or high, making the analysis even more challenging. To address these challenges, we propose a novel sparse multivariate regression framework that simultaneously handles missing values and performs model estimation. The joint distribution of predictors and responses is assumed to follow a multivariate normal distribution, and the lasso penalties are applied to both the regression coefficients and the inverse covariance matrix. We demonstrate the effectiveness of our method through numerical experiments on both simulated and real datasets.