# Orthogonal Projection Loss

arXiv: https://arxiv.org/abs/2103.14021 · HTML: https://mlsanity.com/paper/2103.14021/

## Key facts

- Cited by 44 papers in the map (data through 14 Oct 2026).
- Builds on 51 of its 71 references that are in the map.
- First posted on arXiv 25 Mar 2021.

## Builds on (most cited first, 30 of 51)

- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (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)
- [Learning Deep Features for Discriminative Localization](https://mlsanity.com/paper/1512.04150/) (2015)
- [Barlow Twins: Self-Supervised Learning via Redundancy Reduction](https://mlsanity.com/paper/2103.03230/) (2021)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Reliable evaluation of adversarial robustness with an ensemble of diverse parameter-free attacks](https://mlsanity.com/paper/2003.01690/) (2020)
- [Class-Balanced Loss Based on Effective Number of Samples](https://mlsanity.com/paper/1901.05555/) (2019)
- [How transferable are features in deep neural networks?](https://mlsanity.com/paper/1411.1792/) (2014)
- [Learning to Compare: Relation Network for Few-Shot Learning](https://mlsanity.com/paper/1711.06025/) (2017)
- [Generalized Cross Entropy Loss for Training Deep Neural Networks with Noisy Labels](https://mlsanity.com/paper/1805.07836/) (2018)
- [Deeper, Broader and Artier Domain Generalization](https://mlsanity.com/paper/1710.03077/) (2017)
- [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)
- [Meta-Learning with Differentiable Convex Optimization](https://mlsanity.com/paper/1904.03758/) (2019)
- [Domain Generalization by Solving Jigsaw Puzzles](https://mlsanity.com/paper/1903.06864/) (2019)
- [Robust Loss Functions under Label Noise for Deep Neural Networks](https://mlsanity.com/paper/1712.09482/) (2017)
- [Rethinking Few-Shot Image Classification: a Good Embedding Is All You Need?](https://mlsanity.com/paper/2003.11539/) (2020)
- [Bag of Tricks for Image Classification with Convolutional Neural Networks](https://mlsanity.com/paper/1812.01187/) (2018)
- [Self-Challenging Improves Cross-Domain Generalization](https://mlsanity.com/paper/2007.02454/) (2020)
- [Domain Generalization via Model-Agnostic Learning of Semantic Features](https://mlsanity.com/paper/1910.13580/) (2019)
- [Using Pre-Training Can Improve Model Robustness and Uncertainty](https://mlsanity.com/paper/1901.09960/) (2019)
- [Meta-Dataset: A Dataset of Datasets for Learning to Learn from Few Examples](https://mlsanity.com/paper/1903.03096/) (2019)
- [Large-Margin Softmax Loss for Convolutional Neural Networks](https://mlsanity.com/paper/1612.02295/) (2016)
- [Few-Shot Learning via Embedding Adaptation with Set-to-Set Functions](https://mlsanity.com/paper/1812.03664/) (2018)
- [NormFace: L2 Hypersphere Embedding for Face Verification](https://mlsanity.com/paper/1704.06369/) (2017)
- [Cost Sensitive Learning of Deep Feature Representations from Imbalanced Data](https://mlsanity.com/paper/1508.03422/) (2015)
- [Fast and Flexible Multi-Task Classification Using Conditional Neural Adaptive Processes](https://mlsanity.com/paper/1906.07697/) (2019)

## Built on by (most cited first, 30 of 44)

- [Forward Compatible Few-Shot Class-Incremental Learning](https://mlsanity.com/paper/2203.06953/) (2022)
- [The Principle of Diversity: Training Stronger Vision Transformers Calls for Reducing All Levels of Redundancy](https://mlsanity.com/paper/2203.06345/) (2022)
- [Auxiliary Losses for Learning Generalizable Concept-based Models](https://mlsanity.com/paper/2311.11108/) (2023)
- [Fusion and Orthogonal Projection for Improved Face-Voice Association](https://mlsanity.com/paper/2112.10483/) (2021)
- [Maximum Class Separation as Inductive Bias in One Matrix](https://mlsanity.com/paper/2206.08704/) (2022)
- [Object-Aware Domain Generalization for Object Detection](https://mlsanity.com/paper/2312.12133/) (2023)
- [Benchmarking and Building Long-Context Retrieval Models with LoCo and M2-BERT](https://mlsanity.com/paper/2402.07440/) (2024)
- [EmoSphere++: Emotion-Controllable Zero-Shot Text-to-Speech via Emotion-Adaptive Spherical Vector](https://mlsanity.com/paper/2411.02625/) (2024)
- [Conditional Mutual Information Constrained Deep Learning for Classification](https://mlsanity.com/paper/2309.09123/) (2023)
- [Concept Bottleneck Language Models For protein design](https://mlsanity.com/paper/2411.06090/) (2024)
- [Single-branch Network for Multimodal Training](https://mlsanity.com/paper/2303.06129/) (2023)
- [Enhancing Feature Diversity Boosts Channel-Adaptive Vision Transformers](https://mlsanity.com/paper/2405.16419/) (2024)
- [SCoRe: Submodular Combinatorial Representation Learning](https://mlsanity.com/paper/2310.00165/) (2023)
- [SEAM: Searching Transferable Mixed-Precision Quantization Policy through Large Margin Regularization](https://mlsanity.com/paper/2302.06845/) (2023)
- [SynFacePAD 2023: Competition on Face Presentation Attack Detection Based on Privacy-aware Synthetic Training Data](https://mlsanity.com/paper/2311.05336/) (2023)
- [DiEmo-TTS: Disentangled Emotion Representations via Self-Supervised Distillation for Cross-Speaker Emotion Transfer in Text-to-Speech](https://mlsanity.com/paper/2505.19687/) (2025)
- [Differentiable Projection for Constrained Deep Learning](https://mlsanity.com/paper/2111.10785/) (2021)
- [Distortion-Disentangled Contrastive Learning](https://mlsanity.com/paper/2303.05066/) (2023)
- [Navigating Data Heterogeneity in Federated Learning A Semi-Supervised Federated Object Detection](https://mlsanity.com/paper/2310.17097/) (2023)
- [Attention Guided Cosine Margin For Overcoming Class-Imbalance in Few-Shot Road Object Detection](https://mlsanity.com/paper/2111.06639/) (2021)
- [Learning Branched Fusion and Orthogonal Projection for Face-Voice Association](https://mlsanity.com/paper/2208.10238/) (2022)
- [12 mJ per Class On-Device Online Few-Shot Class-Incremental Learning](https://mlsanity.com/paper/2403.07851/) (2024)
- [Meta Guided Metric Learner for Overcoming Class Confusion in Few-Shot Road Object Detection](https://mlsanity.com/paper/2110.15074/) (2021)
- [DiCLET-TTS: Diffusion Model based Cross-lingual Emotion Transfer for Text-to-Speech -- A Study between English and Mandarin](https://mlsanity.com/paper/2309.00883/) (2023)
- [Predefined Prototypes for Intra-Class Separation and Disentanglement](https://mlsanity.com/paper/2406.16145/) (2024)
- [HyperSpaceX: Radial and Angular Exploration of HyperSpherical Dimensions](https://mlsanity.com/paper/2408.02494/) (2024)
- [Contrastive Consolidation of Top-Down Modulations Achieves Sparsely Supervised Continual Learning](https://mlsanity.com/paper/2505.14125/) (2025)
- [Deconfounding Causal Inference through Two-Branch Framework with Early-Forking for Sensor-Based Cross-Domain Activity Recognition](https://mlsanity.com/paper/2507.03898/) (2025)
- [Hypertext Entity Extraction in Webpage](https://mlsanity.com/paper/2403.01698/) (2024)
- [Enhancing Fine-grained Object Detection in Aerial Images via Orthogonal Mapping](https://mlsanity.com/paper/2407.17738/) (2024)

## Abstract (arXiv)

Deep neural networks have achieved remarkable performance on a range of classification tasks, with softmax cross-entropy (CE) loss emerging as the de-facto objective function. The CE loss encourages features of a class to have a higher projection score on the true class-vector compared to the negative classes. However, this is a relative constraint and does not explicitly force different class features to be well-separated. Motivated by the observation that ground-truth class representations in CE loss are orthogonal (one-hot encoded vectors), we develop a novel loss function termed `Orthogonal Projection Loss' (OPL) which imposes orthogonality in the feature space. OPL augments the properties of CE loss and directly enforces inter-class separation alongside intra-class clustering in the feature space through orthogonality constraints on the mini-batch level. As compared to other alternatives of CE, OPL offers unique advantages e.g., no additional learnable parameters, does not require careful negative mining and is not sensitive to the batch size. Given the plug-and-play nature of OPL, we evaluate it on a diverse range of tasks including image recognition (CIFAR-100), large-scale classification (ImageNet), domain generalization (PACS) and few-shot learning (miniImageNet, CIFAR-FS, tiered-ImageNet and Meta-dataset) and demonstrate its effectiveness across the board. Furthermore, OPL offers better robustness against practical nuisances such as adversarial attacks and label noise. Code is available at: https://github.com/kahnchana/opl.