High-Dimensional Tensor Discriminant Analysis: Low-Rank Discriminant Structure, Representation Synergy, and Theoretical Guarantees

Elynn Chen, Yuefeng Han, Jiayu Li

Abstract:

High-dimensional tensor-valued predictors arise in modern applications, increasingly
as learned representations from neural networks. Existing tensor classification methods
rely on sparsity or Tucker structures and often lack theoretical guarantees. Motivated
by empirical evidence that discriminative signals concentrate along a few multilinear
components, we introduce CP low-rank structure for the discriminant tensor, a modeling perspective not previously explored. Under a Tensor Gaussian Mixture Model, we
propose high-dimensional CP low-rank Tensor Discriminant Analysis (CP-TDA) with
Randomized Composite PCA (rc-PCA) initialization, that is essential for handling
dependent and anisotropic noise under weaker signal strength and incoherence conditions, followed by iterative refinement algorithm. We establish global convergence and
minimax-optimal misclassification rates.
To handle tensor data deviating from tensor normality, we develop the first semiparametric tensor discriminant model, in which learned tensor representations are mapped
via deep generative models into a latent space tailored for CP-TDA. Misclassification
risk decomposes into representation, approximation, and estimation errors. Numerical studies and real data analysis on graph classification demonstrate substantial gains
over existing tensor classifiers and state-of-the-art graph neural networks, particularly
in high-dimensional, small-sample regimes.