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

TL;DR

This paper introduces a dual-channel neural network for tensor data that captures multiway structures, provides finite-sample risk bounds, and develops distribution-free inference and model selection procedures.

Contribution

It proposes a novel dual-channel tensor neural network framework with theoretical guarantees and distribution-free inference methods for tensor structure selection.

Findings

Achieves competitive predictive accuracy in simulations and real data.

Provides finite-sample risk bounds decomposed into approximation, core, and refinement terms.

Develops a distribution-free conformal ROC procedure and structure selector for tensors.

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.