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

arXiv: https://arxiv.org/abs/2609.40131 · HTML: https://mlsanity.com/paper/2609.40131/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 28 references that are in the map.
- First posted on arXiv 30 Sep 2026.

## Builds on (most cited first, 9 of 9)

- [Prototypical Networks for Few-shot Learning](https://mlsanity.com/paper/1703.05175/) (2017)
- [Logic Tensor Networks](https://mlsanity.com/paper/2012.13635/) (2020)
- [From Statistical Relational to Neurosymbolic Artificial Intelligence: a Survey](https://mlsanity.com/paper/2108.11451/) (2021)
- [Towards Data-and Knowledge-Driven Artificial Intelligence: A Survey on Neuro-Symbolic Computing](https://mlsanity.com/paper/2210.15889/) (2022)
- [Neuro-Symbolic Continual Learning: Knowledge, Reasoning Shortcuts and Concept Rehearsal](https://mlsanity.com/paper/2302.01242/) (2023)
- [Neurosymbolic AI for Reasoning over Knowledge Graphs: A Survey](https://mlsanity.com/paper/2302.07200/) (2023)
- [Neuro-symbolic Learning Yielding Logical Constraints](https://mlsanity.com/paper/2410.20957/) (2024)
- [Tensor-Based Classifiers for Hyperspectral Data Analysis](https://mlsanity.com/paper/1709.08164/) (2017)
- [Rank-R FNN: A Tensor-Based Learning Model for High-Order Data Classification](https://mlsanity.com/paper/2104.05048/) (2021)

## Abstract (arXiv)

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.