# Improvising the Learning of Neural Networks on Hyperspherical Manifold

arXiv: https://arxiv.org/abs/2109.14746 · HTML: https://mlsanity.com/paper/2109.14746/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 24 references that are in the map.
- First posted on arXiv 29 Sep 2021.

## Builds on (most cited first, 14 of 14)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Supervised Contrastive Learning](https://mlsanity.com/paper/2004.11362/) (2020)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Visualizing and Understanding Convolutional Networks](https://mlsanity.com/paper/1311.2901/) (2013)
- [Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere](https://mlsanity.com/paper/2005.10242/) (2020)
- [On the Continuity of Rotation Representations in Neural Networks](https://mlsanity.com/paper/1812.07035/) (2018)
- [CosFace: Large Margin Cosine Loss for Deep Face Recognition](https://mlsanity.com/paper/1801.09414/) (2018)
- [SphereFace: Deep Hypersphere Embedding for Face Recognition](https://mlsanity.com/paper/1704.08063/) (2017)
- [Large-Margin Softmax Loss for Convolutional Neural Networks](https://mlsanity.com/paper/1612.02295/) (2016)
- [Hyperspherical Prototype Networks](https://mlsanity.com/paper/1901.10514/) (2019)
- [Deep Hyperspherical Learning](https://mlsanity.com/paper/1711.03189/) (2017)
- [BroadFace: Looking at Tens of Thousands of People at Once for Face Recognition](https://mlsanity.com/paper/2008.06674/) (2020)
- [SPNet: Deep 3D Object Classification and Retrieval using Stereographic Projection](https://mlsanity.com/paper/1811.01571/) (2018)

## Abstract (arXiv)

The impact of convolution neural networks (CNNs) in the supervised settings provided tremendous increment in performance. The representations learned from CNN's operated on hyperspherical manifold led to insightful outcomes in face recognition, face identification, and other supervised tasks. A broad range of activation functions were developed with hypersphere intuition which performs superior to softmax in euclidean space. The main motive of this research is to provide insights. First, the stereographic projection is implied to transform data from Euclidean space (ℝn) to hyperspherical manifold (mathbbSn) to analyze the performance of angular margin losses. Secondly, proving theoretically and practically that decision boundaries constructed on hypersphere using stereographic projection obliges the learning of neural networks. Experiments have demonstrated that applying stereographic projection on existing state-of-the-art angular margin objective functions improved performance for standard image classification data sets (CIFAR-10,100). Further, we ran our experiments on malaria-thin blood smear images, resulting in effective outcomes. The code is publicly available at:https://github.com/barulalithb/stereo-angular-margin.