# Collision Cross-entropy for Soft Class Labels and Deep Clustering

arXiv: https://arxiv.org/abs/2303.07321 · HTML: https://mlsanity.com/paper/2303.07321/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 40 references that are in the map.
- First posted on arXiv 13 Mar 2023.

## Builds on (most cited first, 17 of 17)

- [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)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [An overview of gradient descent optimization algorithms](https://mlsanity.com/paper/1609.04747/) (2016)
- [When Does Label Smoothing Help?](https://mlsanity.com/paper/1906.02629/) (2019)
- [Making Deep Neural Networks Robust to Label Noise: a Loss Correction Approach](https://mlsanity.com/paper/1609.03683/) (2016)
- [Learning from Noisy Labels with Deep Neural Networks: A Survey](https://mlsanity.com/paper/2007.08199/) (2020)
- [Self-labelling via simultaneous clustering and representation learning](https://mlsanity.com/paper/1911.05371/) (2019)
- [Regularizing Neural Networks by Penalizing Confident Output Distributions](https://mlsanity.com/paper/1701.06548/) (2017)
- [Invariant Information Clustering for Unsupervised Image Classification and Segmentation](https://mlsanity.com/paper/1807.06653/) (2018)
- [SCAN: Learning to Classify Images without Labels](https://mlsanity.com/paper/2005.12320/) (2020)
- [Joint Optimization Framework for Learning with Noisy Labels](https://mlsanity.com/paper/1803.11364/) (2018)
- [Learning Discrete Representations via Information Maximizing Self-Augmented Training](https://mlsanity.com/paper/1702.08720/) (2017)
- [Deep Clustering via Joint Convolutional Autoencoder Embedding and Relative Entropy Minimization](https://mlsanity.com/paper/1704.06327/) (2017)
- [Unsupervised and Semi-supervised Learning with Categorical Generative Adversarial Networks](https://mlsanity.com/paper/1511.06390/) (2015)
- [Deep clustering: On the link between discriminative models and K-means](https://mlsanity.com/paper/1810.04246/) (2018)

## Abstract (arXiv)

We propose "collision cross-entropy" as a robust alternative to Shannon's cross-entropy (CE) loss when class labels are represented by soft categorical distributions y. In general, soft labels can naturally represent ambiguous targets in classification. They are particularly relevant for self-labeled clustering methods, where latent pseudo-labels are jointly estimated with the model parameters and uncertainty is prevalent. In case of soft labels, Shannon's CE teaches the model predictions to reproduce the uncertainty in each training example, which inhibits the model's ability to learn and generalize from these examples. As an alternative loss, we propose the negative log of "collision probability" that maximizes the chance of equality between two random variables, predicted class and unknown true class. We show that it has the properties of a generalized CE. The proposed collision CE agrees with Shannon's CE for one-hot labels, but the training from soft labels differs. For example, unlike Shannon's CE, data points where y is a uniform distribution have zero contribution to the training. Collision CE significantly improves classification supervised by soft uncertain targets. Unlike Shannon's, collision CE is symmetric for y and network predictions, which is particularly relevant when both distributions are estimated in the context of self-labeled clustering. Focusing on discriminative deep clustering where self-labeling and entropy-based losses are dominant, we show that the use of collision CE improves the state-of-the-art. We also derive an efficient EM algorithm that significantly speeds up the pseudo-label estimation with collision CE.