Partial Modal Tensor Regression

Overview & Synopsis

With the rapid development of modern scientific research, tensor data have become increasingly prevalent across a wide range of fields. How to effectively characterize the relationship between tensor predictors and a response while preserving the intrinsic structure of tensor data has therefore become an important problem in high-dimensional statistical analysis. Existing tensor regression methods mainly focus on conditional means or conditional quantiles. When the response distribution is asymmetric, heavy-tailed, or contaminated by outlying observations, the conditional modal provides a robust and interpretable alternative characterization. In this paper, we propose a partial modal tensor regression (PMTR) method. The proposed method constructs the partial modal tensor covariance, which incorporates information on the conditional modal to guide tensor dimension reduction, and subsequently establishes a modal tensor regression model in the reduced tensor space. This strategy alleviates the dimensionality burden caused by direct vectorization while preserving the intrinsic structural information of tensor predictors. Under suitable regularity conditions, we establish the consistency and asymptotic properties of the proposed estimators and employ a data-driven dimension-selection procedure to determine the effective dimension along each tensor mode.