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

Eftychios Protopapadakis, Konstantinos Makantasis, Konstantinos M. Giannoutakis

arXiv:2609.40131v1cs.LGcs.CV

TL;DR

Rank-R representations constrain tensor parameterization but do not explicitly arrange classes in the learned representation. This paper adds a differentiable class-level prototype relation to Rank-R learning; under spatial leakage control, the reported Macro-F1 changes vary by dataset, including gains and a decline.

  • Problem

    Rank-R parameter efficiency constrains representation but does not explicitly constrain class geometry in the learned latent space.

    Source: §1 · p. 2

  • Method

    The framework imposes an explicit class-level relation on grounded class symbols within a Rank-R representation.

    Source: §6.4 · p. 30

  • Results

    Under spatial leakage control, full NS improves architecture-averaged Macro-F1 on Botswana by +8.82 pp, while it changes on Salinas by -0.62 pp.

    Source: §7 · pp. 31–32

  • Takeaways & Limitations

    Prototype-rule regularization can complement Rank-R learning primarily by shaping representations, without supporting a universal neurosymbolic advantage.

    Source: §7 · pp. 31–32

    A prototype per class favors compact class geometry and may be inappropriate when that geometry does not fit the data.

    Source: §6.5 · p. 31

Abstract

from arXiv · show

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.

1. Introduction

Rank-R models reduce tensor parameterization but leave learned class geometry unconstrained; this work adds a differentiable prototype rule and evaluates its contribution separately from inference fusion. The study also distinguishes conventional stratified evaluation from spatially separated testing and examines label scarcity.

1. Introduction

  • Rank-R parameter efficiency constrains representation rather than how classes are arranged in the learned latent space.The work frames an explicit prototype relation as potentially complementary to the low-rank tensor constraint.Source: §1 · pp. 2–3
  • A differentiable prototype-rule regularizer is integrated into a CP-constrained Rank-R neural representation.Training-time representation shaping and optional prototype evidence at inference are treated as separate mechanisms.Source: §1 · p. 3, §1.1 · p. 4
  • The novelty claim is deliberately narrow: prior Rank-R FNN studies had not investigated explicit differentiable class rules as an additional training constraint.The paper presents this as a controlled characterization, not a state-of-the-art hyperspectral-classification benchmark.Source: §1 · p. 3

1.1. Contributions and research questions

  • The evaluation separates conventional stratified folds from spatially separated folds to address potentially optimistic patch-based results when neighboring samples cross splits.The authors keep evidence from the two protocols separate.Source: §1 · p. 3, §1.1 · p. 4
  • A dedicated spatial label-scarcity study tests seven per-class budgets with five support realizations.It examines how the effect depends on supervision, dataset, and Rank-R architecture.Source: §1.1 · p. 4
  • The research questions distinguish overall performance change, training regularization versus inference fusion, evaluation protocol, label scarcity, and dataset and architecture dependence.The comparisons retain the same Rank-R backbone when isolating training-time prototype regularization from inference-time fusion.Source: §1.1 · p. 4

2. Related Work

Related work presents Rank-R tensor networks as parameter-efficient representations that leave class geometry unconstrained, and prototype or neurosymbolic methods as sources of explicit class-level structure. The paper combines these ideas through a differentiable relation while preserving the Rank-R architecture.

2. Related Work

  • The two research lines address different aspects: Rank-R methods parameterize high-order inputs, whereas prototype and neurosymbolic methods impose explicit class-level structure.The paper positions its approach as a minimal combination of these lines.Source: §2 · p. 5

2.1. Rank-R tensor neural networks

  • Rank-R tensor networks preserve high-order input structure through CP factors, while the learned representation’s class geometry remains unconstrained.The present work differs by changing the learning objective with an explicit class-level constraint.Source: §2.1 · p. 5

2.2. Neurosymbolic and prototype-based learning

  • Neurosymbolic methods make declarative relations differentiable for joint optimization, while prototype approaches relate embeddings to class representatives.Here, prototypes ground a training constraint rather than a meta-learning episode, and prototype-based inference remains optional.Source: §2.2 · p. 6

2.3. Positioning of the present work

  • The proposed framework uses a minimal differentiable class-level relation to constrain representation geometry while retaining the Rank-R architecture.This supports a controlled evaluation of the added inductive bias.Source: §2.3 · p. 7

3. Proposed Neurosymbolic Rank-R Framework

The framework adds a differentiable prototype rule to a CP-constrained Rank-R representation, shaping class geometry during training and optionally using prototype evidence at inference.

3. Proposed Neurosymbolic Rank-R Framework

  • The framework pairs a CP-constrained Rank-R backbone with prototype-rule regularization and compares neural-only, regularization-only, and regularization-plus-inference-fusion variants.The three variants separate the effects of training-time regularization and inference-time prototype fusion.Source: §3 · pp. 7–8

3.1. Rank-R backbone and latent representation

  • Each hidden unit replaces its dense input-tensor weight with a rank-R CP representation, producing a latent vector that feeds neural class logits.The input tensor is transformed into an embedding by the hidden layer.Source: §3.1 · pp. 8–9

3.2. Prototype-rule neurosymbolic regularization

  • Class prototypes start as the mean embeddings of labeled training samples, then become trainable parameters updated jointly with the Rank-R model after rule activation.The initialization is not permanently recomputed.Source: §3.2 · p. 9
  • For each labeled sample, the rule uses cosine-based distances to attract its embedding to its class prototype while keeping the nearest competitor at least margin m farther away.The prototype-rule constraint is combined with ordinary neural cross-entropy after a warm-up.Source: §3.2 · pp. 9–10

3.3. Training and inference variants

  • RankR uses cross-entropy alone, RankR-NS-RegOnly evaluates the neurosymbolically trained checkpoint with the neural classifier, and RankR-NS adds prototype evidence at inference.The two neurosymbolic variants use the same trained checkpoint, isolating whether prototype-derived logits alter the final decision.Source: §3.3 · pp. 10–11

4. Experimental Methodology

The experimental methodology evaluates Rank-R models on four hyperspectral scenes under fixed architecture and training controls, using stratified and spatial protocols plus a spatial-only label-scarcity study. It assesses class-balanced Macro-F1 with paired comparisons and accounts for the limited number of spatial folds.

4.1. Datasets and input construction

  • Each labeled center is represented by a per-patch-normalized 5×5 neighborhood retaining available spectral channels across four benchmark scenes.The scenes are Pavia University, Indian Pines, Salinas, and Botswana.Source: §4.1 · p. 12

4.2. Architecture grid and training controls

  • The fixed architecture grid crosses ranks R ∈ {3, 5} with hidden dimensions H ∈ {50, 75}, while neurosymbolic training activates its rule after 20 warm-up epochs.Validation Macro-F1 for ordinary neural logits selects checkpoints, including for NS-trained models, so prototype fusion does not affect selection.Source: §4.2 · p. 13

4.3. Spatial and stratified evaluation protocols

  • The primary leakage-aware protocol separates train, validation, and test regions with a four-pixel dead-zone buffer; for 5 × 5 inputs, it prevents patch-footprint overlap and reduces spatial dependence.Source: §4.3 · p. 14

4.4. Label-scarcity protocol

  • The scarcity study uses spatial separation and a fixed 150-update optimization budget as per-class support size increases.Because the update budget stays fixed, the number of passes through the support set is greater at small K than at large K.Source: §4.4 · p. 16
  • Five support seeds generate nested support sets, and K = 1 is excluded because its class prototype equals its sole support embedding.Larger K extends rather than replaces the support ordering for each seed, reducing unrelated resampling as K changes.Source: §4.4 · pp. 16–17
  • The label-scarcity protocol centers on an aggregate class-level relation.Source: §4.4 · p. 17

4.5. Evaluation and statistical analysis

  • Macro-F1 is primary, averaging class-specific F1 scores across classes with ground-truth support; overall accuracy is secondary, and comparisons are paired by split and architecture.Source: §4.5 · p. 17
  • For seven-fold stratification, Friedman tests compare methods, followed when significant by paired Wilcoxon tests with Holm correction; paired direction and effect size are also summarized.Source: §4.5 · p. 17
  • Spatial analyses interpret paired magnitudes and directions rather than rank-based tests because only two or three folds are available per dataset.Source: §4.5 · p. 17
  • Cross-dataset scarcity summaries average within dataset and architecture, then weight the four datasets equally to avoid domination by larger scenes.Source: §4.5 · p. 18
  • Scarcity sensitivity estimates gain slopes against log2 K after averaging support-seed repetitions within spatial fold and architecture; Holm-corrected Wilcoxon tests assess slopes against zero.Source: §4.5 · p. 18
  • Mixed-effects intervals are excluded because singular covariance and a non-positive-definite Hessian made estimates unstable; conclusions rely on paired gains and fold-level slope sensitivity.Source: §4.5 · p. 18

5. Results

Results examine prototype-rule effects across datasets, Rank-R architectures, evaluation protocols, and label budgets, with gains usually positive but varying across settings. Training-time regularization accounts for most spatial gains, while inference fusion is small and dataset dependent; no general label-budget trend is established at the conventional 0.05 level.

5.3. Scarcity impact

  • Across scarcity budgets, Botswana consistently gains +1.10 to +2.92 pp with RankR-NS, while Salinas is negative at six of seven budgets.Pavia University is positive from K = 2 through K = 10, neutral at K = 20 (+0.03 pp), and negative at K = 15 (-0.20 pp); Indian Pines ranges from -0.27 to +1.02 pp depending on K.Source: §5.3 · pp. 23–24

5.2. Architecture dependence

  • Spatial architecture effects reverse on Pavia University: regularization changes are -3.56 and -1.71 pp for R = 3, but +3.60 and +8.20 pp for R = 5.For R = 5, H = 75, the fusion contribution is significantly negative; overall, the effects interact with architecture capacity and structure rather than depending only on dataset.Source: §5.2 · pp. 21–22, Figure 5 · p. 24

5.3. Scarcity impact

  • Across four architectures and tested label budgets, RankR-NS gains over RankR are positive but non-monotonic, ranging from +0.21 pp at K = 7 to +0.97 pp at K = 5.Fusion contribution is modest, ranging from −0.03 pp at K = 7 to +0.35 pp at K = 5.Source: §5.3 · p. 23
  • For RankR-NS versus RankR, 22 of 32 architecture-by-spatial-fold slopes are negative (10 positive) when gain is regressed on log2 K, indicating larger gains when labels are scarcer.Mean slopes are -0.137, -0.039, -0.009, and -0.156 pp per doubling of K for R3-H50, R3-H75, R5-H50, and R5-H75, respectively.Source: §5.3 · p. 25
  • The experiments do not establish a general label-budget trend at the conventional 0.05 level.Source: §5.3 · p. 25
  • The same architecture can change direction across datasets and budgets, with no single low-K regime.Figure 7 makes this heterogeneity explicit.Source: §5.3 · p. 25
  • The full-neurosymbolic Macro-F1 gain over RankR is usually positive across label budgets and Rank-R architectures.Figure 6 summarizes equal-dataset-weighted gains.Source: §5.3 · p. 26, Figure 6 · p. 26

6. Discussion

The discussion qualifies prototype-rule benefits as conditional on dataset, architecture, and evaluation protocol, with training-time regularization accounting for most of the effect and label scarcity showing no monotonic response. It frames the method as a limited class-relation constraint, notes study limitations, and explains that the conservative inferential unit better matches the spatial generalization question despite sacrificing nominal sample size.

6.2. Training-time regularization is the main mechanism

  • Training-time regularization accounts for most of the spatial difference from RankR; inference fusion adds less than one Macro-F1 point and can reduce performance.Fusion is negative on Pavia University and Salinas in the spatial evaluation, and negative across all seven Pavia folds under stratified evaluation.Source: §6.2 · p. 28

6.3. Label scarcity does not produce a simple monotonic response

  • The cross-dataset gain is positive at every tested K, but the aggregate scarcity curve is non-monotonic and no architecture-specific slope survives Holm correction.Although 22 of 32 fold-level slopes favor a larger advantage at lower budgets, the evidence does not support a general claim that scarcity increases the benefit.Source: §6.3 · p. 29

6.4. Why call the mechanism neurosymbolic?

  • The neurosymbolic claim is deliberately modest: a differentiable, declarative class relation constrains learned embeddings, but the prototypes are numerical parameters, not discrete reasoning or external knowledge.Class symbols are minimally grounded in labeled embeddings, and the prototypes may drift with the representation.Source: §6.4 · p. 30

6.5. Limitations and implications

  • Conclusions are limited by one primary model seed, few independent spatial folds, one prototype per class, fixed neurosymbolic hyperparameters, label-derived grounding, and coverage of only four datasets and four architectures.Repeated support-set realizations do not replace independent spatial replications; a single prototype may also be unsuitable for multimodal classes.Source: §6.5 · pp. 30–31
  • The conservative inferential unit sacrifices nominal sample size to better match the spatial generalization question.Source: §6.5 · p. 31

7. Conclusion

Under spatial leakage control, full neurosymbolic inference improves architecture-averaged Macro-F1 on three benchmarks but decreases it on Salinas. Most positive effect comes from training-time prototype regularization; inference fusion is small and dataset dependent.

7. Conclusion

  • Under spatial leakage control, full NS changes architecture-averaged Macro-F1 by +8.82 pp on Botswana, +5.49 pp on Indian Pines, +1.59 pp on Pavia University, and -0.62 pp on Salinas.The regularization-only ablation preserves most of the positive effect.Source: §7 · pp. 31–32
  • Prototype-rule regularization accounts for most of the positive effect, while inference fusion contributes small, dataset-dependent changes and can be harmful.The conclusion separates the training-time regularization effect from inference-time fusion.Source: §7 · p. 32
  • Stratified evaluation confirms heterogeneous outcomes and corrected evidence of a full-NS degradation on Salinas.Source: §7 · p. 32
  • The label-scarcity study finds positive average NS gains across tested budgets, but gains are nonmonotonic and architecture-wise slope tests do not survive Holm correction.Although 22 of 32 fold-level slopes point toward a larger advantage at lower K, none of the architecture-wise slope tests remains significant after correction.Source: §7 · p. 32
  • The authors conclude that prototype-rule regularization complements Rank-R's structural bias primarily by shaping the learned representation.Source: §7 · pp. 31–32
  • The study does not support a universal neurosymbolic advantage; utility depends on dataset, architecture, and evaluation regime.Source: §7 · pp. 31–32
  • Future work should test multi-prototype or structured expert-grounded rules with more independent spatial repetitions and neural seeds.Source: §7 · p. 32
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