# Unveiling the Dynamics of Information Interplay in Supervised Learning

arXiv: https://arxiv.org/abs/2406.03999 · HTML: https://mlsanity.com/paper/2406.03999/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 42 references that are in the map.
- First posted on arXiv 6 Jun 2024.

## Builds on (most cited first, 30 of 36)

- [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)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [Fast R-CNN](https://mlsanity.com/paper/1504.08083/) (2015)
- [FixMatch: Simplifying Semi-Supervised Learning with Consistency and Confidence](https://mlsanity.com/paper/2001.07685/) (2020)
- [Learning both Weights and Connections for Efficient Neural Networks](https://mlsanity.com/paper/1506.02626/) (2015)
- [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)
- [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)
- [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)
- [SoftMatch: Addressing the Quantity-Quality Trade-off in Semi-supervised Learning](https://mlsanity.com/paper/2301.10921/) (2023)
- [Are All Losses Created Equal: A Neural Collapse Perspective](https://mlsanity.com/paper/2210.02192/) (2022)
- [Information Theory with Kernel Methods](https://mlsanity.com/paper/2202.08545/) (2022)
- [Diff-eRank: A Novel Rank-Based Metric for Evaluating Large Language Models](https://mlsanity.com/paper/2401.17139/) (2024)
- [DiME: Maximizing Mutual Information by a Difference of Matrix-Based Entropies](https://mlsanity.com/paper/2301.08164/) (2023)
- [Matrix Information Theory for Self-Supervised Learning](https://mlsanity.com/paper/2305.17326/) (2023)

## Built on by (most cited first, 3 of 3)

- [Beyond Progress Measures: Theoretical Insights into the Mechanism of Grokking](https://mlsanity.com/paper/2504.03162/) (2025)
- [Gradient-Weight Alignment as a Train-Time Proxy for Generalization in Classification Tasks](https://mlsanity.com/paper/2510.25480/) (2025)
- [Neural Collapse by Design: Learning Class Prototypes on the Hypersphere](https://mlsanity.com/paper/2605.20302/) (2026)

## Abstract (arXiv)

In this paper, we use matrix information theory as an analytical tool to analyze the dynamics of the information interplay between data representations and classification head vectors in the supervised learning process. Specifically, inspired by the theory of Neural Collapse, we introduce matrix mutual information ratio (MIR) and matrix entropy difference ratio (HDR) to assess the interactions of data representation and class classification heads in supervised learning, and we determine the theoretical optimal values for MIR and HDR when Neural Collapse happens. Our experiments show that MIR and HDR can effectively explain many phenomena occurring in neural networks, for example, the standard supervised training dynamics, linear mode connectivity, and the performance of label smoothing and pruning. Additionally, we use MIR and HDR to gain insights into the dynamics of grokking, which is an intriguing phenomenon observed in supervised training, where the model demonstrates generalization capabilities long after it has learned to fit the training data. Furthermore, we introduce MIR and HDR as loss terms in supervised and semi-supervised learning to optimize the information interactions among samples and classification heads. The empirical results provide evidence of the method's effectiveness, demonstrating that the utilization of MIR and HDR not only aids in comprehending the dynamics throughout the training process but can also enhances the training procedure itself.