# k* Distribution: Evaluating the Latent Space of Deep Neural Networks using Local Neighborhood Analysis

arXiv: https://arxiv.org/abs/2312.04024 · HTML: https://mlsanity.com/paper/2312.04024/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 80 references that are in the map.
- First posted on arXiv 7 Dec 2023.

## Builds on (most cited first, 30 of 38)

- [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)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Grad-CAM: Visual Explanations from Deep Networks via Gradient-based Localization](https://mlsanity.com/paper/1610.02391/) (2016)
- [EfficientNet: Rethinking Model Scaling for Convolutional Neural Networks](https://mlsanity.com/paper/1905.11946/) (2019)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Aggregated Residual Transformations for Deep Neural Networks](https://mlsanity.com/paper/1611.05431/) (2016)
- [Visualizing and Understanding Convolutional Networks](https://mlsanity.com/paper/1311.2901/) (2013)
- [ImageNet-trained CNNs are biased towards texture; increasing shape bias improves accuracy and robustness](https://mlsanity.com/paper/1811.12231/) (2018)
- [Robustness May Be at Odds with Accuracy](https://mlsanity.com/paper/1805.12152/) (2018)
- [Speech Commands: A Dataset for Limited-Vocabulary Speech Recognition](https://mlsanity.com/paper/1804.03209/) (2018)
- [Understanding Deep Image Representations by Inverting Them](https://mlsanity.com/paper/1412.0035/) (2014)
- [A Multiscale Visualization of Attention in the Transformer Model](https://mlsanity.com/paper/1906.05714/) (2019)
- [Generalisation in humans and deep neural networks](https://mlsanity.com/paper/1808.08750/) (2018)
- [Object Detectors Emerge in Deep Scene CNNs](https://mlsanity.com/paper/1412.6856/) (2014)
- [Neighbor2Neighbor: Self-Supervised Denoising from Single Noisy Images](https://mlsanity.com/paper/2101.02824/) (2021)
- [Understanding and Mitigating the Tradeoff Between Robustness and Accuracy](https://mlsanity.com/paper/2002.10716/) (2020)
- [MASSIVE: A 1M-Example Multilingual Natural Language Understanding Dataset with 51 Typologically-Diverse Languages](https://mlsanity.com/paper/2204.08582/) (2022)
- [Identifying and Controlling Important Neurons in Neural Machine Translation](https://mlsanity.com/paper/1811.01157/) (2018)
- [Neural Nearest Neighbors Networks](https://mlsanity.com/paper/1810.12575/) (2018)
- [Visualizing Large-scale and High-dimensional Data](https://mlsanity.com/paper/1602.00370/) (2016)
- [Summit: Scaling Deep Learning Interpretability by Visualizing Activation and Attribution Summarizations](https://mlsanity.com/paper/1904.02323/) (2019)
- [Seq2Seq-Vis: A Visual Debugging Tool for Sequence-to-Sequence Models](https://mlsanity.com/paper/1804.09299/) (2018)
- [TriMap: Large-scale Dimensionality Reduction Using Triplets](https://mlsanity.com/paper/1910.00204/) (2019)
- [ActiVis: Visual Exploration of Industry-Scale Deep Neural Network Models](https://mlsanity.com/paper/1704.01942/) (2017)
- [Discovering Latent Concepts Learned in BERT](https://mlsanity.com/paper/2205.07237/) (2022)
- [Embedding Projector: Interactive Visualization and Interpretation of Embeddings](https://mlsanity.com/paper/1611.05469/) (2016)
- [Embedding Comparator: Visualizing Differences in Global Structure and Local Neighborhoods via Small Multiples](https://mlsanity.com/paper/1912.04853/) (2019)

## Built on by (most cited first, 2 of 2)

- [Linking Robustness and Generalization: A k* Distribution Analysis of Concept Clustering in Latent Space for Vision Models](https://mlsanity.com/paper/2408.09065/) (2024)
- [Polymorphic Combinatorial Frameworks (PCF): Guiding the Design of Mathematically-Grounded, Adaptive AI Agents](https://mlsanity.com/paper/2508.01581/) (2025)

## Abstract (arXiv)

Most examinations of neural networks' learned latent spaces typically employ dimensionality reduction techniques such as t-SNE or UMAP. These methods distort the local neighborhood in the visualization, making it hard to distinguish the structure of a subset of samples in the latent space. In response to this challenge, we introduce the {k*~distribution} and its corresponding visualization technique This method uses local neighborhood analysis to guarantee the preservation of the structure of sample distributions for individual classes within the subset of the learned latent space. This facilitates easy comparison of different k*~distributions, enabling analysis of how various classes are processed by the same neural network. Our study reveals three distinct distributions of samples within the learned latent space subset: a) Fractured, b) Overlapped, and c) Clustered, providing a more profound understanding of existing contemporary visualizations. Experiments show that the distribution of samples within the network's learned latent space significantly varies depending on the class. Furthermore, we illustrate that our analysis can be applied to explore the latent space of diverse neural network architectures, various layers within neural networks, transformations applied to input samples, and the distribution of training and testing data for neural networks. Thus, the k* distribution should aid in visualizing the structure inside neural networks and further foster their understanding. Project Website is available online at https://shashankkotyan.github.io/k-Distribution/.