# Deep Simplex Classifier for Maximizing the Margin in Both Euclidean and Angular Spaces

arXiv: https://arxiv.org/abs/2212.11747 · HTML: https://mlsanity.com/paper/2212.11747/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 52 references that are in the map.
- First posted on arXiv 22 Dec 2022.

## Builds on (most cited first, 23 of 23)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [FaceNet: A Unified Embedding for Face Recognition and Clustering](https://mlsanity.com/paper/1503.03832/) (2015)
- [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)
- [Prevalence of Neural Collapse during the terminal phase of deep learning training](https://mlsanity.com/paper/2008.08186/) (2020)
- [MS-Celeb-1M: A Dataset and Benchmark for Large-Scale Face Recognition](https://mlsanity.com/paper/1607.08221/) (2016)
- [Towards Open Set Deep Networks](https://mlsanity.com/paper/1511.06233/) (2015)
- [Deep metric learning using Triplet network](https://mlsanity.com/paper/1412.6622/) (2014)
- [Recent Advances in Open Set Recognition: A Survey](https://mlsanity.com/paper/1811.08581/) (2018)
- [Large-Margin Softmax Loss for Convolutional Neural Networks](https://mlsanity.com/paper/1612.02295/) (2016)
- [Cross-Age LFW: A Database for Studying Cross-Age Face Recognition in Unconstrained Environments](https://mlsanity.com/paper/1708.08197/) (2017)
- [Reducing Network Agnostophobia](https://mlsanity.com/paper/1811.04110/) (2018)
- [Classification-Reconstruction Learning for Open-Set Recognition](https://mlsanity.com/paper/1812.04246/) (2018)
- [C2AE: Class Conditioned Auto-Encoder for Open-set Recognition](https://mlsanity.com/paper/1904.01198/) (2019)
- [Learning Open Set Network with Discriminative Reciprocal Points](https://mlsanity.com/paper/2011.00178/) (2020)
- [Range Loss for Deep Face Recognition with Long-tail](https://mlsanity.com/paper/1611.08976/) (2016)
- [Learning towards Minimum Hyperspherical Energy](https://mlsanity.com/paper/1805.09298/) (2018)
- [Learning with Hyperspherical Uniformity](https://mlsanity.com/paper/2103.01649/) (2021)
- [A Theoretically Sound Upper Bound on the Triplet Loss for Improving the Efficiency of Deep Distance Metric Learning](https://mlsanity.com/paper/1904.08720/) (2019)
- [Regularizing Neural Networks via Minimizing Hyperspherical Energy](https://mlsanity.com/paper/1906.04892/) (2019)
- [Contrastive-center loss for deep neural networks](https://mlsanity.com/paper/1707.07391/) (2017)
- [Deep Compact Polyhedral Conic Classifier for Open and Closed Set Recognition](https://mlsanity.com/paper/2102.12570/) (2021)

## Built on by (most cited first, 2 of 2)

- [Enhanced Long-Tailed Recognition with Contrastive CutMix Augmentation](https://mlsanity.com/paper/2407.04911/) (2024)
- [MASIL: Towards Maximum Separable Class Representation for Few Shot Class Incremental Learning](https://mlsanity.com/paper/2304.05362/) (2023)

## Abstract (arXiv)

The classification loss functions used in deep neural network classifiers can be grouped into two categories based on maximizing the margin in either Euclidean or angular spaces. Euclidean distances between sample vectors are used during classification for the methods maximizing the margin in Euclidean spaces whereas the Cosine similarity distance is used during the testing stage for the methods maximizing margin in the angular spaces. This paper introduces a novel classification loss that maximizes the margin in both the Euclidean and angular spaces at the same time. This way, the Euclidean and Cosine distances will produce similar and consistent results and complement each other, which will in turn improve the accuracies. The proposed loss function enforces the samples of classes to cluster around the centers that represent them. The centers approximating classes are chosen from the boundary of a hypersphere, and the pairwise distances between class centers are always equivalent. This restriction corresponds to choosing centers from the vertices of a regular simplex. There is not any hyperparameter that must be set by the user in the proposed loss function, therefore the use of the proposed method is extremely easy for classical classification problems. Moreover, since the class samples are compactly clustered around their corresponding means, the proposed classifier is also very suitable for open set recognition problems where test samples can come from the unknown classes that are not seen in the training phase. Experimental studies show that the proposed method achieves the state-of-the-art accuracies on open set recognition despite its simplicity.