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

Eftychios Protopapadakis; Konstantinos Makantasis; Konstantinos M. Giannoutakis

cs.LGSep 30, 2026

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

Authors: Eftychios Protopapadakis, Konstantinos Makantasis, Konstantinos M. Giannoutakis

Organizations: Department of Applied Informatics, University of Macedonia, 156 Egnatia Street, 54636 Thessaloniki, Greece · Department of Artificial Intelligence, Faculty of Information & Communication Technology, University of Malta, Msida, Malta

Abstract

Rank-constrained tensor neural networks reduce the parameterization of high-order inputs, but they do not explicitly constrain class geometry in the learned representation. This study investigates whether a differentiable prototype-rule can provide a complementary inductive bias for Rank-R tensor learning under limited supervision. The proposed framework augments the Rank-R objective with prototype-based regularization and optionally fuses prototype evidence with neural logits at inference. Four hyperspectral benchmarks are evaluated with four Rank-R configurations under both seven-fold stratification and spatially separated folds that mitigate leakage; a separate spatial study varies the class support budget from 2 to 20 samples. Under spatial evaluation, full neurosymbolic inference changes Macro-F1 score by +8.82 percentage points on Botswana, +5.49 on Indian Pines, +1.59 on Pavia University, and -0.62 on Salinas. Most of the benefit arises from training-time regularization, whereas inference fusion is small and dataset dependent.

Figures & tables

Approach	Tensor structure	Explicit relation	Training constraint	Prototype inference
Rank-R FNN	✓	–	–	–
Prototype-based learning	varies	–	varies	varies
Differentiable constraint-based NS	varies	✓	✓	varies
Present framework	✓	✓	✓	optional

Table 1: Conceptual positioning of the proposed mechanism. The table compares mechanisms rather than predictive performance.

Figure 1: High-level overview of the proposed neurosymbolic Rank-R framework. Hyperspectral input patches are mapped to a latent representation through the CP-constrained Rank-R backbone. The learned embedding can be further shaped by the prototype-rule constraint, which introduces class-prototype attraction and inter-class separation during training. The resulting representation supports three matched variants: the purely neural RankR baseline, RankR-NS-RegOnly using prototype-rule regularization during training, and RankR-NS additionally incorporating prototype evidence during inference.

Method	CE	Rule regularization	Prototype fusion	Scientific role
RankR	✓	–	–	Purely neural Rank-R baseline
RankR-NS-RegOnly	✓	✓	–	Training-time rule effect
RankR-NS	✓	✓	✓	Regularization plus inference fusion

Table 2: Experimental variants and mechanistic interpretation.

Dataset	Spatial size	Bands	Classes	Labeled pixels	Patch
Pavia University	$610\times 340$	103	9	42,776	$5\times 5$
Indian Pines	$145\times 145$	200	16	10,249	$5\times 5$
Salinas	$512\times 217$	204	16	54,129	$5\times 5$
Botswana	$1476\times 256$	145	14	3,248	$5\times 5$

Table 3: Hyperspectral datasets used in the experiments. “Bands” denotes the usable channels in the corrected data used by the experiments.

Panel of Figure 2 — Figure 2: Spatial split configurations used for leakage-controlled evaluation: (a) Botswana, (b) Indian Pines, (c) Pavia University, and (d) Salinas. Blue and green indicate the selected training and validation samples, respectively, drawn from spatially separated eligible regions. Red denotes all valid labeled samples within the spatially held-out test block. Yellow indicates the spatial dead zone introduced to prevent patch overlap between split roles. Light-gray pixels correspond to labeled samples not selected for training, validation, or testing in the illustrated fold, whereas white denotes unlabeled background. Owing to the highly elongated geometry of the Botswana scene, panel (a) shows an enlarged central crop for visual clarity; the split itself is constructed on the complete scene.

Protocol	Dataset	RankR	RankR-NS-RegOnly	RankR-NS
Spatial	Pavia University	$62.57\pm 15.28$	$64.20\pm 18.15$	$64.16\pm 17.96$
	Indian Pines	$24.91\pm 2.26$	$30.19\pm 0.72$	$30.40\pm 0.02$
	Salinas	$58.04\pm 9.24$	$58.18\pm 9.15$	$57.42\pm 8.36$
	Botswana	$48.64\pm 5.21$	$56.58\pm 1.02$	$57.46\pm 2.88$
Stratified	Pavia University	$68.04\pm 1.36$	$69.41\pm 1.50$	$69.23\pm 1.61$
	Indian Pines	$51.77\pm 1.68$	$52.31\pm 1.08$	$51.82\pm 0.90$

Table 4: Absolute dataset-level Macro-F1 (mean

\pm

SD, percentage points). For each method, architecture-specific scores are first averaged within each matched fold and the resulting fold-level values are then summarized across folds.

Protocol	Dataset	RankR-NS-RegOnly $-$ RankR	RankR-NS $-$ RankR	RankR-NS $-$ RankR-NS-RegOnly
Spatial	Pavia University	+1.63	+1.59	-0.05
	Indian Pines	+5.28	+5.49	+0.21
	Salinas	+0.13	-0.62	-0.76
	Botswana	+7.94	+8.82	+0.88
Stratified	Pavia University	+1.36	+1.18	-0.18 ∗
	Indian Pines	+0.54	+0.05	-0.49

Table 5: Dataset-level paired change in Macro-F1 (percentage points). Architecture scores are averaged within matched folds before contrasts are formed. Positive values favor the method named first. An asterisk (*) denotes a Holm-corrected post-hoc

p<0.05

in the seven-fold stratified analysis. No inferential tests are reported for the two/three-fold spatial protocol.

Figure 3: Paired dataset-level Macro-F1 changes under the spatially separated protocol. The small number of spatial folds is treated descriptively; the figure emphasizes effect direction and magnitude rather than fold-level significance.

Figure 4: Paired dataset-level Macro-F1 changes under seven-fold stratified evaluation. Post-hoc testing is performed only when the dataset-level Friedman omnibus test is significant.

Panel of Figure 5 — Figure 5: Architecture-level directional Macro-F1 effects for spatial (left) and stratified (right) evaluation. The heatmaps show that architecture averaging can conceal changes in both magnitude and direction.

$K$	2	3	5	7	10	15	20
RankR-NS $-$ RankR	0.53	0.56	0.97	0.21	0.61	0.46	0.49
RankR-NS-RegOnly $-$ RankR	0.31	0.43	0.62	0.24	0.38	0.39	0.42

Table 6: Equal-dataset-weighted paired Macro-F1 gain over RankR across the label budgets. Values are percentage points and are averaged over the four Rank-R architectures after dataset-level aggregation.

Figure 6: Equal-dataset-weighted full-neurosymbolic Macro-F1 gain over RankR across label budgets and Rank-R architectures. The overall gain is usually positive but not monotonic in $K$ .

Panel of Figure 7 — Figure 7: Dataset-specific RankR-NS gain over RankR across label budgets. The principal result is heterogeneity rather than a universal scarcity curve.

Explore similar work

Dual-Channel Tensor Neural Networks: Finite-Sample Theory and Conformal Structure Selection
May 18, 2026Elynn Chen, Jiayu Li, Zheshi Zheng +1Conformal Selection Finite-Sample
New York University · University of Michigan · Duke University
Low-Rank Tensor Completion Based on Fractional Regularization with Ky Fan p-k Norm
Jun 17, 2026Shan Fan, Feng Zhang, Jianjun Wang +2Tensor Completion Low-Rank Structure
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China · School of Mathematical Sciences/Research Center for Image and Vision Computing, University of Electronic Science and Technology of China, Chengdu 611731, China · Faculty of Computer Science and Control Engineering, Shenzhen University of Advanced Technology, Shenzhen 518055, China
EinSort: Sorting is All We Need for Tensorizing LLM
Jun 7, 2026Toshiaki Koike-Akino, Jing Liu, Ye WangTensor Networks Low-Rank Structure
Mitsubishi Electric Research Laboratories (MERL), 20 Broadway, Cambridge, MA 02139, USA.

May 18, 2026, stat.MLJ/K move · Enter open · S save

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

Elynn Chen, Jiayu Li, Zheshi Zheng, Jian Pei

New York University · University of Michigan · Duke University

Conformal Selection Finite-Sample Theory

Open paper AskPDF arXiv

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.

Abstract

Explore similar work

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

Low-Rank Tensor Completion Based on Fractional Regularization with Ky Fan p-k Norm

EinSort: Sorting is All We Need for Tensorizing LLM