Abstract: Multivariate varying-coefficient models are popular statistical tools for analyzing the relationship between multiple responses and covariates. Nevertheless, estimating large numbers of coefficient functions is challenging, especially with limited samples. In this work, we propose a reduced-dimension model based on the Tucker decomposition that unifies several existing models. In addition, we use sparse predictor effects, in the sense that only a few predictors are related to the responses, to achieve an interpretable model and sufficiently reduce the number of unknown functions to be estimated. These dimension-reduction and sparsity considerations are integrated into a penalized least squares problem on the constraint domain of third-order tensors. To compute the proposed estimator, we propose a block updating algorithm based on the alternating direction method of multipliers and manifold optimization. We also establish the oracle inequality for the prediction risk of the proposed estimator. A real data set from the Framingham Heart Study is used to demonstrate the good predictive performance of the proposed method.

Key words and phrases: Dimensionality reduction, group Lasso, polynomial splines, sparsity, Tucker low rank.