# Assessing Neural Network Representations During Training Using Noise-Resilient Diffusion Spectral Entropy

arXiv: https://arxiv.org/abs/2312.04823 · HTML: https://mlsanity.com/paper/2312.04823/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 42 references that are in the map.
- First posted on arXiv 4 Dec 2023.

## Builds on (most cited first, 13 of 13)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Swin Transformer: Hierarchical Vision Transformer using Shifted Windows](https://mlsanity.com/paper/2103.14030/) (2021)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Aggregated Residual Transformations for Deep Neural Networks](https://mlsanity.com/paper/1611.05431/) (2016)
- [Deep Learning and the Information Bottleneck Principle](https://mlsanity.com/paper/1503.02406/) (2015)
- [MINE: Mutual Information Neural Estimation](https://mlsanity.com/paper/1801.04062/) (2018)
- [XCiT: Cross-Covariance Image Transformers](https://mlsanity.com/paper/2106.09681/) (2021)
- [Efficient Estimation of Mutual Information for Strongly Dependent Variables](https://mlsanity.com/paper/1411.2003/) (2014)
- [Estimating Mixture Entropy with Pairwise Distances](https://mlsanity.com/paper/1706.02419/) (2017)
- [Time-inhomogeneous diffusion geometry and topology](https://mlsanity.com/paper/2203.14860/) (2022)

## Built on by (most cited first, 13 of 13)

- [Approximating mutual information of high-dimensional variables using learned representations](https://mlsanity.com/paper/2409.02732/) (2024)
- [ImageFlowNet: Forecasting Multiscale Image-Level Trajectories of Disease Progression with Irregularly-Sampled Longitudinal Medical Images](https://mlsanity.com/paper/2406.14794/) (2024)
- [Not All Directions Matter: Towards Structured and Task-Aware Low-Rank Model Adaptation](https://mlsanity.com/paper/2603.14228/) (2026)
- [Geometry-Aware Generative Autoencoders for Warped Riemannian Metric Learning and Generative Modeling on Data Manifolds](https://mlsanity.com/paper/2410.12779/) (2024)
- [Staying VIGILant: Mitigating Visual Laziness via Counterfactual Visual Alignment in MLLMs](https://mlsanity.com/paper/2606.26387/) (2026)
- [DiffKillR: Killing and Recreating Diffeomorphisms for Cell Annotation in Dense Microscopy Images](https://mlsanity.com/paper/2410.03058/) (2024)
- [Multimodal Conditional Information Bottleneck for Generalizable AI-Generated Image Detection](https://mlsanity.com/paper/2505.15217/) (2025)
- [Hyperedge Representations with Hypergraph Wavelets: Applications to Spatial Transcriptomics](https://mlsanity.com/paper/2409.09469/) (2024)
- [Dispersion Loss Counteracts Embedding Condensation and Improves Generalization in Small Language Models](https://mlsanity.com/paper/2602.00217/) (2026)
- [Diffusion Operator Geometry of Feedforward Representations](https://mlsanity.com/paper/2605.01107/) (2026)
- [RNAGenScape: Property-Guided, Optimized Generation of mRNA Sequences with Manifold Langevin Dynamics](https://mlsanity.com/paper/2510.24736/) (2025)
- [A Spectral Decomposition Framework for Multiscale Nonlinear Dimensionality Reduction](https://mlsanity.com/paper/2604.02535/) (2026)
- [Auditing Finite-Lag Statistics of Recurrent Representations](https://mlsanity.com/paper/2607.01746/) (2026)

## Abstract (arXiv)

Entropy and mutual information in neural networks provide rich information on the learning process, but they have proven difficult to compute reliably in high dimensions. Indeed, in noisy and high-dimensional data, traditional estimates in ambient dimensions approach a fixed entropy and are prohibitively hard to compute. To address these issues, we leverage data geometry to access the underlying manifold and reliably compute these information-theoretic measures. Specifically, we define diffusion spectral entropy (DSE) in neural representations of a dataset as well as diffusion spectral mutual information (DSMI) between different variables representing data. First, we show that they form noise-resistant measures of intrinsic dimensionality and relationship strength in high-dimensional simulated data that outperform classic Shannon entropy, nonparametric estimation, and mutual information neural estimation (MINE). We then study the evolution of representations in classification networks with supervised learning, self-supervision, or overfitting. We observe that (1) DSE of neural representations increases during training; (2) DSMI with the class label increases during generalizable learning but stays stagnant during overfitting; (3) DSMI with the input signal shows differing trends: on MNIST it increases, while on CIFAR-10 and STL-10 it decreases. Finally, we show that DSE can be used to guide better network initialization and that DSMI can be used to predict downstream classification accuracy across 962 models on ImageNet. The official implementation is available at https://github.com/ChenLiu-1996/DiffusionSpectralEntropy.