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Multivariate regression models are widely used in various fields such as biology and finance. In this paper, we focus on two key challenges: (a) When should we favor a multivariate model over a series of univariate models; (b) If the numbers of responses and predictors are allowed to greatly exceed the sample size, how to reduce the computational cost and provide precise estimation. The proposed method, Interaction Pursuit Biconvex Optimization (IPBO), explores the regression relationship allowing the predictors and responses derived from different multivariate normal distributions with general covariance matrices. In practice, the correlation structures within are complex and interact on each other based on the regression function. The proposed method solves this problem by building a structured sparsity penalty to encourages the shared structure between the network and the regression coefficients. We prove theoretical results under interpretable conditions, and provide an efficient algorithm to compute the estimator. Simulation studies and real data examples compare the proposed method with several existing methods, indicating that IPBO works well.