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

2025-12-12

AI Summary

To address challenges in high-dimensional tensor classification—including difficulty modeling low-rank discriminative structures, lack of theoretical guarantees, and performance degradation under small-sample regimes—this paper proposes the first semi-parametric tensor discriminant analysis framework grounded in CP low-rank structure. Methodologically, it introduces CP decomposition to explicitly capture multilinear sparsity of discriminative signals; integrates randomized composite PCA initialization with an iterative refinement algorithm; and ensures robustness against anisotropic and dependent noise. Theoretically, it establishes global convergence guarantees and derives minimax-optimal misclassification rate bounds. Empirically, on real-world tasks such as graph classification—particularly in high-dimensional, low-sample settings—the method reduces misclassification rates by up to 32% compared to state-of-the-art tensor classifiers and graph neural networks.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 ( extsc{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.

Problem

• Develops CP low-rank tensor discriminant analysis for high-dimensional tensor classification
• Establishes theoretical guarantees for global convergence and optimal misclassification rates
• Proposes semiparametric model integrating deep generative representations for non-normal data

Innovation

• CP low-rank structure for discriminant tensor modeling
• Randomized Composite PCA initialization for noise handling
• Semiparametric model with deep generative latent mapping