Dual-Channel Tensor Neural Networks: Finite-Sample Theory and Conformal Structure Selection

2026-05-18

AI Summary

This work addresses the challenge that existing tensor learning methods struggle to simultaneously capture global low-rank structure and local details—either modeling only a single low-rank form or disrupting intrinsic multidimensional geometry through vectorization. To overcome this limitation, the authors propose a Dual-Channel Tensor Neural Network (DC-TNN) that decomposes the input tensor into a low-rank core and a sparse correction term, processed respectively by coupled neural channels. The framework flexibly accommodates multiple decomposition formats, including CP, Tucker, and tensor train. A novel, distribution-free and finite-sample-efficient tensor structure selection method is introduced, along with a structure-aware conformal ROC inference framework to enable reliable decomposition selection and uncertainty quantification. Experiments demonstrate that the proposed approach achieves high predictive accuracy, well-calibrated uncertainty estimates, and consistent structural recovery on both synthetic and protein data.

Abstract

Tensor-valued data arise naturally in neuroimaging, genomics, climate science, and spatiotemporal networks, where multilinear dependencies across modes carry information that is destroyed under vectorization. Existing approaches either impose a single low-rank structure, which can miss localized signal, or treat the tensor as a long vector, which discards its multiway geometry. We propose a *Dual-Channel Tensor Neural Network* (DC-TNN) that decomposes each tensor input into a low-rank core and a sparse refinement, and processes the two components through coupled neural channels. The framework is structure-agnostic and accommodates CP, Tucker, and tensor-train cores within a single architecture. For estimation, we establish non-asymptotic risk bounds for the DC-TNN estimator that decompose into network approximation, core estimation, and refinement-selection terms, and show that the effective dimension is determined jointly by the core rank and refinement sparsity rather than by the ambient tensor size. For inference, we develop a *structure-aware conformal ROC* procedure that calibrates within the core-refinement latent space and produces ROC and AUC confidence bands with finite-sample, distribution-free coverage. Building on this, we propose a *conformal structure selector* that, to our knowledge, is the *first distribution-free procedure* for choosing among candidate tensor decompositions with finite-sample validity. Simulations and an analysis of a protein dataset demonstrate competitive predictive accuracy, reliable uncertainty quantification, and consistent recovery of the tensor structure.

Problem

• tensor-valued data
• low-rank structure
• multiway geometry
• localized signal
• tensor decomposition

Innovation

• Dual-Channel Tensor Neural Network
• finite-sample theory
• conformal structure selection
• tensor decomposition
• non-asymptotic risk bounds