# von Mises-Fisher Loss: An Exploration of Embedding Geometries for Supervised Learning

arXiv: https://arxiv.org/abs/2103.15718 · HTML: https://mlsanity.com/paper/2103.15718/

## Key facts

- Cited by 42 papers in the map (data through 14 Oct 2026).
- Builds on 53 of its 82 references that are in the map.
- First posted on arXiv 29 Mar 2021.

## Builds on (most cited first, 30 of 53)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Prototypical Networks for Few-shot Learning](https://mlsanity.com/paper/1703.05175/) (2017)
- [FaceNet: A Unified Embedding for Face Recognition and Clustering](https://mlsanity.com/paper/1503.03832/) (2015)
- [Matching Networks for One Shot Learning](https://mlsanity.com/paper/1606.04080/) (2016)
- [Learning to Compare: Relation Network for Few-Shot Learning](https://mlsanity.com/paper/1711.06025/) (2017)
- [CosFace: Large Margin Cosine Loss for Deep Face Recognition](https://mlsanity.com/paper/1801.09414/) (2018)
- [Deep Variational Information Bottleneck](https://mlsanity.com/paper/1612.00410/) (2016)
- [SphereFace: Deep Hypersphere Embedding for Face Recognition](https://mlsanity.com/paper/1704.08063/) (2017)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Deep Metric Learning via Lifted Structured Feature Embedding](https://mlsanity.com/paper/1511.06452/) (2015)
- [Rethinking Few-Shot Image Classification: a Good Embedding Is All You Need?](https://mlsanity.com/paper/2003.11539/) (2020)
- [Circle Loss: A Unified Perspective of Pair Similarity Optimization](https://mlsanity.com/paper/2002.10857/) (2020)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Multi-Similarity Loss with General Pair Weighting for Deep Metric Learning](https://mlsanity.com/paper/1904.06627/) (2019)
- [A Metric Learning Reality Check](https://mlsanity.com/paper/2003.08505/) (2020)
- [Verified Uncertainty Calibration](https://mlsanity.com/paper/1909.10155/) (2019)
- [Sampling Matters in Deep Embedding Learning](https://mlsanity.com/paper/1706.07567/) (2017)
- [Additive Margin Softmax for Face Verification](https://mlsanity.com/paper/1801.05599/) (2018)
- [NormFace: L2 Hypersphere Embedding for Face Verification](https://mlsanity.com/paper/1704.06369/) (2017)
- [No Fuss Distance Metric Learning using Proxies](https://mlsanity.com/paper/1703.07464/) (2017)
- [Hyperbolic Image Embeddings](https://mlsanity.com/paper/1904.02239/) (2019)
- [Hyperspherical Variational Auto-Encoders](https://mlsanity.com/paper/1804.00891/) (2018)
- [Probabilistic Face Embeddings](https://mlsanity.com/paper/1904.09658/) (2019)
- [SoftTriple Loss: Deep Metric Learning Without Triplet Sampling](https://mlsanity.com/paper/1909.05235/) (2019)
- [Learning a Deep Embedding Model for Zero-Shot Learning](https://mlsanity.com/paper/1611.05088/) (2016)
- [Data Uncertainty Learning in Face Recognition](https://mlsanity.com/paper/2003.11339/) (2020)
- [Deep Metric Learning with Angular Loss](https://mlsanity.com/paper/1708.01682/) (2017)
- [ProxyNCA++: Revisiting and Revitalizing Proxy Neighborhood Component Analysis](https://mlsanity.com/paper/2004.01113/) (2020)

## Built on by (most cited first, 30 of 42)

- [Probabilistic Contrastive Learning for Long-Tailed Visual Recognition](https://mlsanity.com/paper/2403.06726/) (2024)
- [Transductive Zero-Shot and Few-Shot CLIP](https://mlsanity.com/paper/2405.18437/) (2024)
- [Probabilistic Contrastive Learning Recovers the Correct Aleatoric Uncertainty of Ambiguous Inputs](https://mlsanity.com/paper/2302.02865/) (2023)
- [Geometric Autoencoders -- What You See is What You Decode](https://mlsanity.com/paper/2306.17638/) (2023)
- [Representation Uncertainty in Self-Supervised Learning as Variational Inference](https://mlsanity.com/paper/2203.11437/) (2022)
- [A Non-isotropic Probabilistic Take on Proxy-based Deep Metric Learning](https://mlsanity.com/paper/2207.03784/) (2022)
- [URL: A Representation Learning Benchmark for Transferable Uncertainty Estimates](https://mlsanity.com/paper/2307.03810/) (2023)
- [Boosting Semi-Supervised Semantic Segmentation with Probabilistic Representations](https://mlsanity.com/paper/2210.14670/) (2022)
- [Seeing Unseen: Discover Novel Biomedical Concepts via Geometry-Constrained Probabilistic Modeling](https://mlsanity.com/paper/2403.01053/) (2024)
- [On the Importance of Embedding Norms in Self-Supervised Learning](https://mlsanity.com/paper/2502.09252/) (2025)
- [Probabilistic Embeddings Revisited](https://mlsanity.com/paper/2202.06768/) (2022)
- [Shedding a PAC-Bayesian Light on Adaptive Sliced-Wasserstein Distances](https://mlsanity.com/paper/2206.03230/) (2022)
- [Introspective Deep Metric Learning](https://mlsanity.com/paper/2309.09982/) (2023)
- [Variational Classification](https://mlsanity.com/paper/2305.10406/) (2023)
- [Factorizers for Distributed Sparse Block Codes](https://mlsanity.com/paper/2303.13957/) (2023)
- [Your Classifier Can Be Secretly a Likelihood-Based OOD Detector](https://mlsanity.com/paper/2408.04851/) (2024)
- [Exploiting the Asymmetric Uncertainty Structure of Pre-trained VLMs on the Unit Hypersphere](https://mlsanity.com/paper/2505.11029/) (2025)
- [Magnitude-aware Probabilistic Speaker Embeddings](https://mlsanity.com/paper/2202.13826/) (2022)
- [ScaleFace: Uncertainty-aware Deep Metric Learning](https://mlsanity.com/paper/2209.01880/) (2022)
- [Few-shot Classification via Ensemble Learning with Multi-Order Statistics](https://mlsanity.com/paper/2305.00454/) (2023)
- [PRCL: Probabilistic Representation Contrastive Learning for Semi-Supervised Semantic Segmentation](https://mlsanity.com/paper/2402.18117/) (2024)
- [Context-Aware Multimodal Pretraining](https://mlsanity.com/paper/2411.15099/) (2024)
- [Harmonizing Geometry and Uncertainty: Diffusion with Hyperspheres](https://mlsanity.com/paper/2506.10576/) (2025)
- [vMFCoOp: Towards Equilibrium on a Unified Hyperspherical Manifold for Prompting Biomedical VLMs](https://mlsanity.com/paper/2511.09540/) (2025)
- [The Unreasonable Effectiveness of Random Target Embeddings for Continuous-Output Neural Machine Translation](https://mlsanity.com/paper/2310.20620/) (2023)
- [Distribution Guidance Network for Weakly Supervised Point Cloud Semantic Segmentation](https://mlsanity.com/paper/2410.08091/) (2024)
- [An Empirical Study on Clustering Pretrained Embeddings: Is Deep Strictly Better?](https://mlsanity.com/paper/2211.05183/) (2022)
- [Dynamic Feature Learning and Matching for Class-Incremental Learning](https://mlsanity.com/paper/2405.08533/) (2024)
- [SphOR: A Representation Learning Perspective on Open-set Recognition for Identifying Unknown Classes in Deep Learning Models](https://mlsanity.com/paper/2503.08049/) (2025)
- [Spherical Cauchy Variational Autoencoders: Heavy Angular Tails and Exact KL Evaluation](https://mlsanity.com/paper/2506.21278/) (2025)

## Abstract (arXiv)

Recent work has argued that classification losses utilizing softmax cross-entropy are superior not only for fixed-set classification tasks, but also by outperforming losses developed specifically for open-set tasks including few-shot learning and retrieval. Softmax classifiers have been studied using different embedding geometries -- Euclidean, hyperbolic, and spherical -- and claims have been made about the superiority of one or another, but they have not been systematically compared with careful controls. We conduct an empirical investigation of embedding geometry on softmax losses for a variety of fixed-set classification and image retrieval tasks. An interesting property observed for the spherical losses lead us to propose a probabilistic classifier based on the von Mises-Fisher distribution, and we show that it is competitive with state-of-the-art methods while producing improved out-of-the-box calibration. We provide guidance regarding the trade-offs between losses and how to choose among them.