# Feature Space Topology Control via Hopkins Loss

arXiv: https://arxiv.org/abs/2509.11154 · HTML: https://mlsanity.com/paper/2509.11154/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 40 references that are in the map.
- First posted on arXiv 14 Sep 2025.

## Builds on (most cited first, 15 of 15)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Focal Frequency Loss for Image Reconstruction and Synthesis](https://mlsanity.com/paper/2012.12821/) (2020)
- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [A Topology Layer for Machine Learning](https://mlsanity.com/paper/1905.12200/) (2019)
- [How Well Generative Adversarial Networks Learn Distributions](https://mlsanity.com/paper/1811.03179/) (2018)
- [Unsupervised Domain Adaptation using Feature-Whitening and Consensus Loss](https://mlsanity.com/paper/1903.03215/) (2019)
- [Rethinking Feature Distribution for Loss Functions in Image Classification](https://mlsanity.com/paper/1803.02988/) (2018)
- [Manifold Learning by Mixture Models of VAEs for Inverse Problems](https://mlsanity.com/paper/2303.15244/) (2023)
- [An Automated Data Mining Framework Using Autoencoders for Feature Extraction and Dimensionality Reduction](https://mlsanity.com/paper/2412.02211/) (2024)
- [Controllable Generative Adversarial Network](https://mlsanity.com/paper/1708.00598/) (2017)
- [A Topological Loss Function: Image Denoising on a Low-Light Dataset](https://mlsanity.com/paper/2208.04573/) (2022)
- [Enhancing the Resilience of Graph Neural Networks to Topological Perturbations in Sparse Graphs](https://mlsanity.com/paper/2406.03097/) (2024)

## Abstract (arXiv)

Feature space topology refers to the organization of samples within the feature space. Modifying this topology can be beneficial in machine learning applications, including dimensionality reduction, generative modeling, transfer learning, and robustness to adversarial attacks. This paper introduces a novel loss function, Hopkins loss, which leverages the Hopkins statistic to enforce a desired feature space topology, which is in contrast to existing topology-related methods that aim to preserve input feature topology. We evaluate the effectiveness of Hopkins loss on speech, text, and image data in two scenarios: classification and dimensionality reduction using nonlinear bottleneck autoencoders. Our experiments show that integrating Hopkins loss into classification or dimensionality reduction has only a small impact on classification performance while providing the benefit of modifying feature topology.