# Topologically Densified Distributions

arXiv: https://arxiv.org/abs/2002.04805 · HTML: https://mlsanity.com/paper/2002.04805/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Feb 2020.

## Built on by (most cited first, 8 of 8)

- [Neural Collapse with Normalized Features: A Geometric Analysis over the Riemannian Manifold](https://mlsanity.com/paper/2209.09211/) (2022)
- [Relative Representations: Topological and Geometric Perspectives](https://mlsanity.com/paper/2409.10967/) (2024)
- [On the Convergence of Optimizing Persistent-Homology-Based Losses](https://mlsanity.com/paper/2206.02946/) (2022)
- [Be Persistent: Towards a Unified Solution for Mitigating Shortcuts in Deep Learning](https://mlsanity.com/paper/2402.11237/) (2024)
- [Deep Invertible Approximation of Topologically Rich Maps between Manifolds](https://mlsanity.com/paper/2210.00577/) (2022)
- [Decorrelating neurons using persistence](https://mlsanity.com/paper/2308.04870/) (2023)
- [Operational Abstractions of Neural Network Concepts via Topological Representations](https://mlsanity.com/paper/2610.05017/) (2026)
- [PLDiv: Measuring Dataset Diversity and Semantic Collapse via Persistence Landscapes](https://mlsanity.com/paper/2610.14031/) (2026)

## Abstract (arXiv)

We study regularization in the context of small sample-size learning with over-parameterized neural networks. Specifically, we shift focus from architectural properties, such as norms on the network weights, to properties of the internal representations before a linear classifier. Specifically, we impose a topological constraint on samples drawn from the probability measure induced in that space. This provably leads to mass concentration effects around the representations of training instances, i.e., a property beneficial for generalization. By leveraging previous work to impose topological constraints in a neural network setting, we provide empirical evidence (across various vision benchmarks) to support our claim for better generalization.