# How deep is deep enough? -- Quantifying class separability in the hidden layers of deep neural networks

arXiv: https://arxiv.org/abs/1811.01753 · HTML: https://mlsanity.com/paper/1811.01753/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Nov 2018.

## Built on by (most cited first, 28 of 28)

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- [Inversion dynamics of class manifolds in deep learning reveals tradeoffs underlying generalisation](https://mlsanity.com/paper/2303.05161/) (2023)
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- [Rethinking U-net Skip Connections for Biomedical Image Segmentation](https://mlsanity.com/paper/2402.08276/) (2024)
- [Analysis of Argument Structure Constructions in a Deep Recurrent Language Model](https://mlsanity.com/paper/2408.03062/) (2024)
- [Analysis of Argument Structure Constructions in the Large Language Model BERT](https://mlsanity.com/paper/2408.04270/) (2024)
- [SCHEME: Scalable Channel Mixer for Vision Transformers](https://mlsanity.com/paper/2312.00412/) (2023)
- [Constructing Enhanced Mutual Information for Online Class-Incremental Learning](https://mlsanity.com/paper/2407.18526/) (2024)
- [A classification performance evaluation measure considering data separability](https://mlsanity.com/paper/2211.05433/) (2022)
- [Hidden Classification Layers: Enhancing linear separability between classes in neural networks layers](https://mlsanity.com/paper/2306.06146/) (2023)
- [Beyond Labels: Advancing Cluster Analysis with the Entropy of Distance Distribution (EDD)](https://mlsanity.com/paper/2311.16621/) (2023)
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- [On Representational Dissociation of Language and Arithmetic in Large Language Models](https://mlsanity.com/paper/2502.11932/) (2025)
- [Certified Topological Interaction in Neural Representations: Exact Tests and the Statistic They Require](https://mlsanity.com/paper/2609.08561/) (2026)
- [Predictive coding and stochastic resonance as fundamental principles of auditory perception](https://mlsanity.com/paper/2204.03354/) (2022)
- [Classifying Overlapping Gaussian Mixtures in High Dimensions: From Optimal Classifiers to Neural Nets](https://mlsanity.com/paper/2405.18427/) (2024)
- [Author-Specific Linguistic Patterns Unveiled: A Deep Learning Study on Word Class Distributions](https://mlsanity.com/paper/2501.10072/) (2025)
- [Improving Fine-Tuning with Latent Cluster Correction](https://mlsanity.com/paper/2501.11919/) (2025)
- [Don't stop me now: How Validation Criteria Affect Checkpoint Selection and Early Stopping](https://mlsanity.com/paper/2602.22107/) (2026)
- [Neuroscience-Inspired Analyses of Visual Interestingness in Multimodal Transformers](https://mlsanity.com/paper/2605.08188/) (2026)

## Abstract (arXiv)

Deep neural networks typically outperform more traditional machine learning models in their ability to classify complex data, and yet is not clear how the individual hidden layers of a deep network contribute to the overall classification performance. We thus introduce a Generalized Discrimination Value (GDV) that measures, in a non-invasive manner, how well different data classes separate in each given network layer. The GDV can be used for the automatic tuning of hyper-parameters, such as the width profile and the total depth of a network. Moreover, the layer-dependent GDV(L) provides new insights into the data transformations that self-organize during training: In the case of multi-layer perceptrons trained with error backpropagation, we find that classification of highly complex data sets requires a temporal em reduction of class separability, marked by a characteristic 'energy barrier' in the initial part of the GDV(L) curve. Even more surprisingly, for a given data set, the GDV(L) is running through a fixed 'master curve', independently from the total number of network layers. Furthermore, applying the GDV to Deep Belief Networks reveals that also unsupervised training with the Contrastive Divergence method can systematically increase class separability over tens of layers, even though the system does not 'know' the desired class labels. These results indicate that the GDV may become a useful tool to open the black box of deep learning.