# Wrapped Cauchy Distributed Angular Softmax for Long-Tailed Visual Recognition

arXiv: https://arxiv.org/abs/2305.18732 · HTML: https://mlsanity.com/paper/2305.18732/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 48 references that are in the map.
- First posted on arXiv 30 May 2023.

## Builds on (most cited first, 25 of 25)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Class-Balanced Loss Based on Effective Number of Samples](https://mlsanity.com/paper/1901.05555/) (2019)
- [Learning Imbalanced Datasets with Label-Distribution-Aware Margin Loss](https://mlsanity.com/paper/1906.07413/) (2019)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [Decoupling Representation and Classifier for Long-Tailed Recognition](https://mlsanity.com/paper/1910.09217/) (2019)
- [SphereFace: Deep Hypersphere Embedding for Face Recognition](https://mlsanity.com/paper/1704.08063/) (2017)
- [Large-Scale Long-Tailed Recognition in an Open World](https://mlsanity.com/paper/1904.05160/) (2019)
- [Deep Long-Tailed Learning: A Survey](https://mlsanity.com/paper/2110.04596/) (2021)
- [Balanced Meta-Softmax for Long-Tailed Visual Recognition](https://mlsanity.com/paper/2007.10740/) (2020)
- [Large-Margin Softmax Loss for Convolutional Neural Networks](https://mlsanity.com/paper/1612.02295/) (2016)
- [Equalization Loss for Long-Tailed Object Recognition](https://mlsanity.com/paper/2003.05176/) (2020)
- [Long-tailed Recognition by Routing Diverse Distribution-Aware Experts](https://mlsanity.com/paper/2010.01809/) (2020)
- [Parametric Contrastive Learning](https://mlsanity.com/paper/2107.12028/) (2021)
- [Free Lunch for Few-shot Learning: Distribution Calibration](https://mlsanity.com/paper/2101.06395/) (2021)
- [Data Uncertainty Learning in Face Recognition](https://mlsanity.com/paper/2003.11339/) (2020)
- [Contrastive Learning based Hybrid Networks for Long-Tailed Image Classification](https://mlsanity.com/paper/2103.14267/) (2021)
- [Deep Imbalanced Learning for Face Recognition and Attribute Prediction](https://mlsanity.com/paper/1806.00194/) (2018)
- [Identifying and Compensating for Feature Deviation in Imbalanced Deep Learning](https://mlsanity.com/paper/2001.01385/) (2020)
- [Exploring Classification Equilibrium in Long-Tailed Object Detection](https://mlsanity.com/paper/2108.07507/) (2021)
- [Adversarial Robustness under Long-Tailed Distribution](https://mlsanity.com/paper/2104.02703/) (2021)
- [Heteroskedastic and Imbalanced Deep Learning with Adaptive Regularization](https://mlsanity.com/paper/2006.15766/) (2020)
- [VideoLT: Large-scale Long-tailed Video Recognition](https://mlsanity.com/paper/2105.02668/) (2021)
- [Long-Tailed Partial Label Learning via Dynamic Rebalancing](https://mlsanity.com/paper/2302.05080/) (2023)

## Built on by (most cited first, 11 of 11)

- [Long-Tail Learning with Foundation Model: Heavy Fine-Tuning Hurts](https://mlsanity.com/paper/2309.10019/) (2023)
- [Balancing the Scales: A Theoretical and Algorithmic Framework for Learning from Imbalanced Data](https://mlsanity.com/paper/2502.10381/) (2025)
- [Latent-based Diffusion Model for Long-tailed Recognition](https://mlsanity.com/paper/2404.04517/) (2024)
- [A Survey of Deep Long-Tail Classification Advancements](https://mlsanity.com/paper/2404.15593/) (2024)
- [Continuous Contrastive Learning for Long-Tailed Semi-Supervised Recognition](https://mlsanity.com/paper/2410.06109/) (2024)
- [Improved Balanced Classification with Theoretically Grounded Loss Functions](https://mlsanity.com/paper/2512.23947/) (2025)
- [Adaptive Parametric Activation: Unifying and Generalising Activation Functions Across Tasks](https://mlsanity.com/paper/2407.08567/) (2024)
- [LIFT+: Lightweight Fine-Tuning for Long-Tail Learning](https://mlsanity.com/paper/2504.13282/) (2025)
- [PROTOCOL: Partial Optimal Transport-enhanced Contrastive Learning for Imbalanced Multi-view Clustering](https://mlsanity.com/paper/2506.12408/) (2025)
- [Optimized Deferral for Imbalanced Settings](https://mlsanity.com/paper/2604.27723/) (2026)
- [Mixture of Balanced Information Bottlenecks for Long-Tailed Visual Recognition](https://mlsanity.com/paper/2509.01804/) (2025)

## Abstract (arXiv)

Addressing imbalanced or long-tailed data is a major challenge in visual recognition tasks due to disparities between training and testing distributions and issues with data noise. We propose the Wrapped Cauchy Distributed Angular Softmax (WCDAS), a novel softmax function that incorporates data-wise Gaussian-based kernels into the angular correlation between feature representations and classifier weights, effectively mitigating noise and sparse sampling concerns. The class-wise distribution of angular representation becomes a sum of these kernels. Our theoretical analysis reveals that the wrapped Cauchy distribution excels the Gaussian distribution in approximating mixed distributions. Additionally, WCDAS uses trainable concentration parameters to dynamically adjust the compactness and margin of each class. Empirical results confirm label-aware behavior in these parameters and demonstrate WCDAS's superiority over other state-of-the-art softmax-based methods in handling long-tailed visual recognition across multiple benchmark datasets. The code is public available.