# Bhargava Cube--Inspired Quadratic Regularization for Structured Neural Embeddings

arXiv: https://arxiv.org/abs/2512.11392 · HTML: https://mlsanity.com/paper/2512.11392/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 30 references that are in the map.
- First posted on arXiv 12 Dec 2025.

## Builds on (most cited first, 13 of 13)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Dynamic Routing Between Capsules](https://mlsanity.com/paper/1710.09829/) (2017)
- [Regularizing Neural Networks by Penalizing Confident Output Distributions](https://mlsanity.com/paper/1701.06548/) (2017)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [TensorLy: Tensor Learning in Python](https://mlsanity.com/paper/1610.09555/) (2016)
- [On the Computational Efficiency of Training Neural Networks](https://mlsanity.com/paper/1410.1141/) (2014)
- [Spectral Learning on Matrices and Tensors](https://mlsanity.com/paper/2004.07984/) (2020)
- [Sparse Unsupervised Capsules Generalize Better](https://mlsanity.com/paper/1804.06094/) (2018)
- [Capsule Networks with Max-Min Normalization](https://mlsanity.com/paper/1903.09662/) (2019)

## Abstract (arXiv)

We present a novel approach to neural representation learning that incorporates algebraic constraints inspired by Bhargava cubes from number theory. Traditional deep learning methods learn representations in unstructured latent spaces lacking interpretability and mathematical consistency. Our framework maps input data to constrained 3-dimensional latent spaces where embeddings are regularized to satisfy learned quadratic relationships derived from Bhargava's combinatorial structures. The architecture employs a differentiable auxiliary loss function operating independently of classification objectives, guiding models toward mathematically structured representations. We evaluate on MNIST, achieving 99.46% accuracy while producing interpretable 3D embeddings that naturally cluster by digit class and satisfy learned quadratic constraints. Unlike existing manifold learning approaches requiring explicit geometric supervision, our method imposes weak algebraic priors through differentiable constraints, ensuring compatibility with standard optimization. This represents the first application of number-theoretic constructs to neural representation learning, establishing a foundation for incorporating structured mathematical priors in neural networks.