Efficient Decision Trees for Tensor Regressions

In recent years, the intersection of tensor data analysis and non-parametric modeling (Guhaniyogi et al., 2017; Papadogeorgou et al., 2021; Wang and Xu, 2024) has garnered considerable interest among mathematicians and statisticians. Non-parametric tensor models have the potential to handle complex multi-dimensional data (Bi et al., 2021) and represent spatial correlation between entries of data. This paper addresses both scalar-on-tensor (i.e., to predict a scalar response based on a tensor input) and tensor-on-tensor (i.e., both the input and output are tensors) non-linear regression problems using recursive partitioning methods, often referred to as tree(-based) models. Supervised learning on tensor data, such as tensor regression, has significant relevance due to the proliferation of multi-dimensional data in modern applications. Tensor data naturally arises in various fields such as imaging (Wang and Xu, 2024), neuroscience (Li et al., 2018), and computer vision (Luo and Ma, 2023), where observations often take the form of multi-way arrays. Traditional regression models typically handle vector inputs and outputs, and thus can fail to capture the structural information embedded within tensor data.