Prototype-Rule Neurosymbolic Regularization for Rank-Constrained Tensor Neural Networks under Label Scarcity

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.

This paper addresses low-rank tensor completion (LRTC) by proposing a novel nonconvex surrogate, namely the ratio of the tensor nuclear norm to the tensor Ky Fan p-k norm (TNPK), to accurately approximate the tensor tubal rank. The TNPK possesses appealing properties, including scale invariance, parameter flexibility, and the existence of closed-form solutions under specific choices of p and k. With specific parameter settings of p and k, it reduces to the ratio of the tensor nuclear norm to the tensor Ky Fan k norm (TNK) or the ratio of the tensor nuclear norm to the tensor Frobenius norm (TNF). We construct a LRTC model and, under the tensor null space property (NSP), prove that low-rank tensors are local minimizers of the proposed model. Moreover, we derive the proximal operator of the Ky Fan p-k inverse-norm and further develop an efficient alternating direction method of multipliers (ADMM) algorithm with guaranteed subsequential convergence under mild conditions. Extensive experiments on synthetic and real-world datasets validate the superior performance of our method against state-of-the-art competitors.

Tensor networks provide efficient representations for compressing large neural networks. By carefully designing shapes and topologies, they can significantly reduce memory and computational costs. However, identifying implicit low-rank structures in large foundation models remains challenging due to their enormous scale and un-structured weight distributions. We propose an adaptive tensorization method that discovers inherent low-rank structure in a target tensor by index ordering. Experiments on weight and KV-cache compression demonstrate improved reconstruction quality compared to baselines.