# Data Representations' Study of Latent Image Manifolds

arXiv: https://arxiv.org/abs/2305.19730 · HTML: https://mlsanity.com/paper/2305.19730/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 62 references that are in the map.
- First posted on arXiv 31 May 2023.

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

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [mixup: Beyond Empirical Risk Minimization](https://mlsanity.com/paper/1710.09412/) (2017)
- [Rich feature hierarchies for accurate object detection and semantic segmentation](https://mlsanity.com/paper/1311.2524/) (2013)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Deep metric learning using Triplet network](https://mlsanity.com/paper/1412.6622/) (2014)
- [Measuring the Intrinsic Dimension of Objective Landscapes](https://mlsanity.com/paper/1804.08838/) (2018)
- [Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers](https://mlsanity.com/paper/1811.04918/) (2018)
- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [Characterizing Adversarial Subspaces Using Local Intrinsic Dimensionality](https://mlsanity.com/paper/1801.02613/) (2018)
- [Intrinsic dimension of data representations in deep neural networks](https://mlsanity.com/paper/1905.12784/) (2019)
- [Exponential expressivity in deep neural networks through transient chaos](https://mlsanity.com/paper/1606.05340/) (2016)
- [Estimating the intrinsic dimension of datasets by a minimal neighborhood information](https://mlsanity.com/paper/1803.06992/) (2018)
- [Latent Space Oddity: on the Curvature of Deep Generative Models](https://mlsanity.com/paper/1710.11379/) (2017)
- [Dimensionality-Driven Learning with Noisy Labels](https://mlsanity.com/paper/1806.02612/) (2018)
- [ReduNet: A White-box Deep Network from the Principle of Maximizing Rate Reduction](https://mlsanity.com/paper/2105.10446/) (2021)
- [The Riemannian Geometry of Deep Generative Models](https://mlsanity.com/paper/1711.08014/) (2017)
- [Intrinsic Dimension, Persistent Homology and Generalization in Neural Networks](https://mlsanity.com/paper/2111.13171/) (2021)
- [Classification and Geometry of General Perceptual Manifolds](https://mlsanity.com/paper/1710.06487/) (2017)
- [On the Intrinsic Dimensionality of Image Representations](https://mlsanity.com/paper/1803.09672/) (2018)
- [Deep Networks and the Multiple Manifold Problem](https://mlsanity.com/paper/2008.11245/) (2020)
- [Learning Flat Latent Manifolds with VAEs](https://mlsanity.com/paper/2002.04881/) (2020)
- [Hierarchical nucleation in deep neural networks](https://mlsanity.com/paper/2007.03506/) (2020)
- [Metrics for Probabilistic Geometries](https://mlsanity.com/paper/1411.7432/) (2014)
- [LDMNet: Low Dimensional Manifold Regularized Neural Networks](https://mlsanity.com/paper/1711.06246/) (2017)
- [Manifold GPLVMs for discovering non-Euclidean latent structure in neural data](https://mlsanity.com/paper/2006.07429/) (2020)
- [Efficient Representation of Low-Dimensional Manifolds using Deep Networks](https://mlsanity.com/paper/1602.04723/) (2016)
- [Curvature-aware Manifold Learning](https://mlsanity.com/paper/1706.07167/) (2017)
- [Curvature-based Comparison of Two Neural Networks](https://mlsanity.com/paper/1801.06801/) (2018)

## Built on by (most cited first, 11 of 11)

- [Utilizing Image Transforms and Diffusion Models for Generative Modeling of Short and Long Time Series](https://mlsanity.com/paper/2410.19538/) (2024)
- [First-Order Manifold Data Augmentation for Regression Learning](https://mlsanity.com/paper/2406.10914/) (2024)
- [Analyzing Deep Transformer Models for Time Series Forecasting via Manifold Learning](https://mlsanity.com/paper/2410.13792/) (2024)
- [Intrinsic Dimension Correlation: uncovering nonlinear connections in multimodal representations](https://mlsanity.com/paper/2406.15812/) (2024)
- [A Multi-Task Learning Approach to Linear Multivariate Forecasting](https://mlsanity.com/paper/2502.03571/) (2025)
- [Investigating the Impact of Hard Samples on Accuracy Reveals In-class Data Imbalance](https://mlsanity.com/paper/2409.14401/) (2024)
- [Curvature Enhanced Data Augmentation for Regression](https://mlsanity.com/paper/2506.06853/) (2025)
- [Unraveling Hidden Representations: A Multi-Modal Layer Analysis for Better Synthetic Content Forensics](https://mlsanity.com/paper/2508.00784/) (2025)
- [Reducing Class Bias In Data-Balanced Datasets Through Hardness-Based Resampling](https://mlsanity.com/paper/2504.07031/) (2025)
- [Privacy-Utility Trade-off in Data Publication: A Bilevel Optimization Framework with Curvature-Guided Perturbation](https://mlsanity.com/paper/2509.02048/) (2025)
- [The Data Manifold under the Microscope](https://mlsanity.com/paper/2606.15760/) (2026)

## Abstract (arXiv)

Deep neural networks have been demonstrated to achieve phenomenal success in many domains, and yet their inner mechanisms are not well understood. In this paper, we investigate the curvature of image manifolds, i.e., the manifold deviation from being flat in its principal directions. We find that state-of-the-art trained convolutional neural networks for image classification have a characteristic curvature profile along layers: an initial steep increase, followed by a long phase of a plateau, and followed by another increase. In contrast, this behavior does not appear in untrained networks in which the curvature flattens. We also show that the curvature gap between the last two layers has a strong correlation with the generalization capability of the network. Moreover, we find that the intrinsic dimension of latent codes is not necessarily indicative of curvature. Finally, we observe that common regularization methods such as mixup yield flatter representations when compared to other methods. Our experiments show consistent results over a variety of deep learning architectures and multiple data sets. Our code is publicly available at https://github.com/azencot-group/CRLM