# Torus embeddings

arXiv: https://arxiv.org/abs/2603.03135 · HTML: https://mlsanity.com/paper/2603.03135/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 29 references that are in the map.
- First posted on arXiv 3 Mar 2026.

## Builds on (most cited first, 16 of 16)

- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [DINOv2: Learning Robust Visual Features without Supervision](https://mlsanity.com/paper/2304.07193/) (2023)
- [QLoRA: Efficient Finetuning of Quantized LLMs](https://mlsanity.com/paper/2305.14314/) (2023)
- [Supervised Contrastive Learning](https://mlsanity.com/paper/2004.11362/) (2020)
- [Quantization and Training of Neural Networks for Efficient Integer-Arithmetic-Only Inference](https://mlsanity.com/paper/1712.05877/) (2017)
- [Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere](https://mlsanity.com/paper/2005.10242/) (2020)
- [XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks](https://mlsanity.com/paper/1603.05279/) (2016)
- [NormFace: L2 Hypersphere Embedding for Face Verification](https://mlsanity.com/paper/1704.06369/) (2017)
- [Efficient Deep Learning: A Survey on Making Deep Learning Models Smaller, Faster, and Better](https://mlsanity.com/paper/2106.08962/) (2021)
- [Microscaling Data Formats for Deep Learning](https://mlsanity.com/paper/2310.10537/) (2023)
- [Spreading vectors for similarity search](https://mlsanity.com/paper/1806.03198/) (2018)
- [nGPT: Normalized Transformer with Representation Learning on the Hypersphere](https://mlsanity.com/paper/2410.01131/) (2024)
- [A Closer Look at Few-shot Classification Again](https://mlsanity.com/paper/2301.12246/) (2023)
- [Regularizing Neural Networks via Minimizing Hyperspherical Energy](https://mlsanity.com/paper/1906.04892/) (2019)
- [Hubs and Hyperspheres: Reducing Hubness and Improving Transductive Few-shot Learning with Hyperspherical Embeddings](https://mlsanity.com/paper/2303.09352/) (2023)
- [Domain-Invariant Representation Learning of Bird Sounds](https://mlsanity.com/paper/2409.08589/) (2024)

## Abstract (arXiv)

Many data representations are vectors of continuous values. In particular, deep learning embeddings are data-driven representations, typically either unconstrained in Euclidean space, or constrained to a hypersphere. These may also be translated into integer representations (quantised) for efficient large-scale use. However, the fundamental (and most efficient) numeric representation in the overwhelming majority of existing computers is integers with overflow -- and vectors of these integers do not correspond to either of these spaces, but instead to the topology of a (hyper)torus. This mismatch can lead to wasted representation capacity. Here we show that common deep learning frameworks can be adapted, quite simply, to create representations with inherent toroidal topology. We investigate two alternative strategies, demonstrating that a normalisation-based strategy leads to training with desirable stability and performance properties, comparable to a standard hyperspherical L2 normalisation. We also demonstrate that a torus embedding maintains desirable quantisation properties. The torus embedding does not outperform hypersphere embeddings in general, but is comparable, and opens the possibility to train deep embeddings which have an extremely simple pathway to efficient `TinyML' embedded implementation.