Tensor Neyman-Pearson Classification: Theory, Algorithms, and Error Control (2512.04583v1)

Abstract: Biochemical discovery increasingly relies on classifying molecular structures when the consequences of different errors are highly asymmetric. In mutagenicity and carcinogenicity, misclassifying a harmful compound as benign can trigger substantial scientific, regulatory, and health risks, whereas false alarms primarily increase laboratory workload. Modern representations transform molecular graphs into persistence image tensors that preserve multiscale geometric and topological structure, yet existing tensor classifiers and deep tensor neural networks provide no finite-sample guarantees on type I error and often exhibit severe error inflation in practice. We develop the first Tensor Neyman-Pearson (Tensor-NP) classification framework that achieves finite-sample control of type I error while exploiting the multi-mode structure of tensor data. Under a tensor-normal mixture model, we derive the oracle NP discriminant, characterize its Tucker low-rank manifold geometry, and establish tensor-specific margin and conditional detection conditions enabling high-probability bounds on excess type II error. We further propose a Discriminant Tensor Iterative Projection estimator and a Tensor-NP Neural Classifier combining deep learning with Tensor-NP umbrella calibration, yielding the first distribution-free NP-valid methods for multiway data. Across four biochemical datasets, Tensor-NP classifiers maintain type I errors at prespecified levels while delivering competitive type II error performance, providing reliable tools for asymmetric-risk decisions with complex molecular tensors.

• The paper introduces a novel Tensor-NP classification framework that achieves finite-sample Type I error control for high-dimensional tensor data.
• It develops two models—T-LDA-NP and T-NN-NP—that combine tensor geometry with deep learning to enhance error control and classification accuracy.
• Extensive theoretical analysis and empirical validations demonstrate robust performance and improved reliability in asymmetric-risk biochemical screening.

Tensor Neyman-Pearson Classification: Theory, Algorithms, and Error Control

Introduction

The paper introduces a novel "Tensor Neyman-Pearson (Tensor-NP) Classification" framework which addresses the complex challenges posed by asymmetric error risks in biochemical data classification. Traditional methods lack finite-sample guarantees for controlling Type I errors in tensor classifiers, especially when dealing with molecular data represented as high-dimensional multiway arrays derived from molecular graphs. In applications such as mutagenicity and carcinogenicity screening, the risk of misclassifying harmful compounds is asymmetrically more critical than false alarms. Modern deep learning and tensor classifiers often fail to provide statistical reliability and finite-sample guarantees crucial for such high-stakes decisions.

Proposed Methods

The Tensor-NP classification framework establishes finite-sample control of Type I errors while leveraging the inherent multi-mode structure of tensor data. The framework is based on a tensor-normal mixture model, where the oracle NP discriminant is characterized by its Tucker low-rank manifold geometry. This geometric characterization enables deriving high-probability bounds on excess Type II error through conditional margin and detection conditions tailored for tensor data.

Tensor-Based Models and Algorithms

Two primary models are highlighted:

1. Tensor Linear Discriminant Analysis (T-LDA-NP): The T-LDA-NP model assumes a tensor-normal distribution for predictors, allowing for a closed-form solution for the oracle classifier under controlled Type I error conditions. The Discriminant Tensor Iterative Projection (DTIP) estimator ensures Tucker low-rank manifold geometry, facilitating finite-sample NP oracle inequalities adapting to tensor dimensions and ranks.
2. Tensor Neural Network NP Classifier (T-NN-NP): This utilizes deep learning integrated with tensor-specific architecture, including tensor contraction layers (TCLs) to maintain multi-mode dependencies. These layers significantly reduce parameters and enhance structural representation while the NP umbrella algorithm guarantees type I error control, making T-NN-NP model-agnostic and distribution-free.

   Figure 1: The architecture diagram of the tensor neural network model in T-NN and T-NN-NP.

Theoretical Foundations

The theoretical analysis substantiates the finite-sample error control as follows:

• Scoring Function Deviation: A high-probability bound for scoring functions ensures that deviations from the oracle NP classifier decrease with increasing sample size.
• Conditional Margin and Detection Conditions: These are proven to hold under tensor settings, which are critical for connecting excess Type II error control with finite-sample guarantees.
• Oracle Inequalities: Together, they ensure the proposed Tensor-NP classifiers maintain type I error below a specified level and diminish excess type II error as sample sizes grow.

  Figure 2: Average Type I Error, Average Type II Error and Accuracy in Example \ref{exp:vary distribution.

Results

Extensive empirical analyses validate the efficacy of Tensor-NP classifiers across synthetic datasets representing tensor data and real-world biochemical datasets like MUTAG, COX2, and BZR:

• Under varying sample sizes and tensor shapes, Tensor-NP classifiers consistently control type I error within target levels, demonstrating adaptability to structural data complexity.
• When compared to traditional neural network models that violate type I error constraints, Tensor-NP frameworks exhibit balanced performance managing asymmetric risks with robust accuracy.

Practical Implications

The Tensor-NP approach greatly impacts asymmetric-risk decision-making scenarios in biochemical applications, promoting reliable classifications in drug discovery pipelines, toxicity screening, and enzyme activity prediction. By uniting tensor structures and NP theoretical guarantees, the framework notably improves reliability and interpretability in high-dimensional data settings that characterize molecular bioscience applications.

Conclusion

Tensor Neyman-Pearson classification represents a significant advance in both deep learning and statistical methodology by delivering reliable type I error control, proving crucial for informed and balanced decisions in asymmetric-risk scenarios. Future work lies in extending tensor NP classification to other structured prediction models ensuring robust asymmetric error control in various scientific domains.