# CCAR: Intrinsic Robustness as an Emergent Geometric Property

arXiv: https://arxiv.org/abs/2604.16861 · HTML: https://mlsanity.com/paper/2604.16861/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 35 references that are in the map.
- First posted on arXiv 18 Apr 2026.

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

- [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)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [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)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Supervised Contrastive Learning](https://mlsanity.com/paper/2004.11362/) (2020)
- [Visualizing and Understanding Convolutional Networks](https://mlsanity.com/paper/1311.2901/) (2013)
- [Shortcut Learning in Deep Neural Networks](https://mlsanity.com/paper/2004.07780/) (2020)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [ImageNet-trained CNNs are biased towards texture; increasing shape bias improves accuracy and robustness](https://mlsanity.com/paper/1811.12231/) (2018)
- [Understanding intermediate layers using linear classifier probes](https://mlsanity.com/paper/1610.01644/) (2016)
- [CosFace: Large Margin Cosine Loss for Deep Face Recognition](https://mlsanity.com/paper/1801.09414/) (2018)
- [Toy Models of Superposition](https://mlsanity.com/paper/2209.10652/) (2022)
- [Adversarial Examples Are Not Bugs, They Are Features](https://mlsanity.com/paper/1905.02175/) (2019)
- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [Prevalence of Neural Collapse during the terminal phase of deep learning training](https://mlsanity.com/paper/2008.08186/) (2020)
- [Network Dissection: Quantifying Interpretability of Deep Visual Representations](https://mlsanity.com/paper/1704.05796/) (2017)
- [DivideMix: Learning with Noisy Labels as Semi-supervised Learning](https://mlsanity.com/paper/2002.07394/) (2020)
- [Robust Loss Functions under Label Noise for Deep Neural Networks](https://mlsanity.com/paper/1712.09482/) (2017)
- [Robustness of classifiers: from adversarial to random noise](https://mlsanity.com/paper/1608.08967/) (2016)
- [Implicit Rank-Minimizing Autoencoder](https://mlsanity.com/paper/2010.00679/) (2020)
- [Non-negative Contrastive Learning](https://mlsanity.com/paper/2403.12459/) (2024)

## Abstract (arXiv)

Standard supervised learning optimizes for predictive accuracy but remains agnostic to the internal geometry of learned features, often yielding representations that are entangled and brittle. We propose Class-Conditional Activation Regularization (CCAR) to explicitly engineer the feature space, imposing a block-diagonal structure via a soft inductive bias. By shaping the latent representation to confine class energy to orthogonal subspaces, we create an intrinsic geometric scaffold that naturally filters noise and adversarial perturbations. We provide theoretical analysis linking this structural constraint to the maximization of the Fisher Discriminant Ratio, establishing a formal connection between geometric disentanglement and algorithmic stability. Empirically, this approach demonstrates that robustness is an emergent property of a well-engineered feature space, significantly outperforming baselines on label noise and input corruption benchmarks.