# Exploring Information-Theoretic Metrics Associated with Neural Collapse in Supervised Training

arXiv: https://arxiv.org/abs/2409.16767 · HTML: https://mlsanity.com/paper/2409.16767/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 42 of its 58 references that are in the map.
- First posted on arXiv 25 Sep 2024.

## Builds on (most cited first, 30 of 42)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Microsoft COCO: Common Objects in Context](https://mlsanity.com/paper/1405.0312/) (2014)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Fast R-CNN](https://mlsanity.com/paper/1504.08083/) (2015)
- [Learning to Prompt for Vision-Language Models](https://mlsanity.com/paper/2109.01134/) (2021)
- [Describing Textures in the Wild](https://mlsanity.com/paper/1311.3618/) (2013)
- [FixMatch: Simplifying Semi-Supervised Learning with Consistency and Confidence](https://mlsanity.com/paper/2001.07685/) (2020)
- [Fine-Grained Visual Classification of Aircraft](https://mlsanity.com/paper/1306.5151/) (2013)
- [EuroSAT: A Novel Dataset and Deep Learning Benchmark for Land Use and Land Cover Classification](https://mlsanity.com/paper/1709.00029/) (2017)
- [Conditional Prompt Learning for Vision-Language Models](https://mlsanity.com/paper/2203.05557/) (2022)
- [MixMatch: A Holistic Approach to Semi-Supervised Learning](https://mlsanity.com/paper/1905.02249/) (2019)
- [Unsupervised Data Augmentation for Consistency Training](https://mlsanity.com/paper/1904.12848/) (2019)
- [Virtual Adversarial Training: A Regularization Method for Supervised and Semi-Supervised Learning](https://mlsanity.com/paper/1704.03976/) (2017)
- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [Prevalence of Neural Collapse during the terminal phase of deep learning training](https://mlsanity.com/paper/2008.08186/) (2020)
- [MaPLe: Multi-modal Prompt Learning](https://mlsanity.com/paper/2210.03117/) (2022)
- [Linear Mode Connectivity and the Lottery Ticket Hypothesis](https://mlsanity.com/paper/1912.05671/) (2019)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [Listen, Attend and Spell](https://mlsanity.com/paper/1508.01211/) (2015)
- [FlexMatch: Boosting Semi-Supervised Learning with Curriculum Pseudo Labeling](https://mlsanity.com/paper/2110.08263/) (2021)
- [Meta Pseudo Labels](https://mlsanity.com/paper/2003.10580/) (2020)
- [ReMixMatch: Semi-Supervised Learning with Distribution Alignment and Augmentation Anchoring](https://mlsanity.com/paper/1911.09785/) (2019)
- [Self-regulating Prompts: Foundational Model Adaptation without Forgetting](https://mlsanity.com/paper/2307.06948/) (2023)
- [Semi-Supervised Learning with Ladder Networks](https://mlsanity.com/paper/1507.02672/) (2015)
- [FreeMatch: Self-adaptive Thresholding for Semi-supervised Learning](https://mlsanity.com/paper/2205.07246/) (2022)
- [Neural Collapse Under MSE Loss: Proximity to and Dynamics on the Central Path](https://mlsanity.com/paper/2106.02073/) (2021)
- [RankMe: Assessing the downstream performance of pretrained self-supervised representations by their rank](https://mlsanity.com/paper/2210.02885/) (2022)
- [Dash: Semi-Supervised Learning with Dynamic Thresholding](https://mlsanity.com/paper/2109.00650/) (2021)

## Built on by (most cited first, 1 of 1)

- [Fine-Grained Generalization via Structuralizing Concept and Feature Space into Commonality, Specificity and Confounding](https://mlsanity.com/paper/2601.03056/) (2026)

## Abstract (arXiv)

In this paper, we introduce matrix entropy as an analytical tool for studying supervised learning, investigating the information content of data representations and classification head vectors, as well as the dynamic interactions between them during the supervised learning process. Our experimental results reveal that matrix entropy effectively captures the variations in information content of data representations and classification head vectors as neural networks approach Neural Collapse during supervised training, while also serving as a robust metric for measuring similarity among data samples. Leveraging this property, we propose Cross-Model Alignment (CMA) loss to optimize the fine-tuning of pretrained models. To characterize the dynamics of neural networks nearing the Neural Collapse state, we introduce two novel metrics: the Matrix Mutual Information Ratio (MIR) and the Matrix Entropy Difference Ratio (HDR), which quantitatively assess the interactions between data representations and classification heads in supervised learning, with theoretical optimal values derived under the Neural Collapse state. Our experiments demonstrate that MIR and HDR effectively explain various phenomena in neural networks, including the dynamics of standard supervised training, linear mode connectivity. Moreover, we use MIR and HDR to analyze the dynamics of grokking, which is a fascinating phenomenon in supervised learning where a model unexpectedly exhibits generalization long after achieving training data fit.