Generalized high-dimensional tensor learning with nuclear norm regularization

Abstract

Generalized linear models (GLMs) are widely used in diverse domains but are limited to vector or matrix data, overlooking modern high-dimensional structures such as videos and color images. To address this gap, the storage of information in multi-way arrays, known as tensors, has become a common practice. Inspired by the proposed tensor singular value decomposition (t-SVD), this paper introduces a generalized tensor regression learning framework that incorporates a tensor nuclear norm regularization. This framework elegantly extends the capabilities of traditional GLMs, which are tailored for vector and matrix data, to handle low-rank, high-dimensional tensors for regression and classification tasks. We also rigorously establish theoretical guarantees, including the convergence rate of the proposed estimator, to ensure statistical validity. Numerical experiments on synthetic datasets confirm its efficiency and superiority over matrix-based methods, while applications to image classification further demonstrate its practical effectiveness.