# Nonparametric Topological Layers in Neural Networks

arXiv: https://arxiv.org/abs/2111.14829 · HTML: https://mlsanity.com/paper/2111.14829/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 25 references that are in the map.
- First posted on arXiv 27 Nov 2021.

## Builds on (most cited first, 12 of 12)

- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [Self-Distillation Amplifies Regularization in Hilbert Space](https://mlsanity.com/paper/2002.05715/) (2020)
- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [Deep Learning with Topological Signatures](https://mlsanity.com/paper/1707.04041/) (2017)
- [Topology of deep neural networks](https://mlsanity.com/paper/2004.06093/) (2020)
- [A Topology Layer for Machine Learning](https://mlsanity.com/paper/1905.12200/) (2019)
- [A Stable Multi-Scale Kernel for Topological Machine Learning](https://mlsanity.com/paper/1412.6821/) (2014)
- [A Topological Regularizer for Classifiers via Persistent Homology](https://mlsanity.com/paper/1806.10714/) (2018)
- [Topological Uncertainty: Monitoring trained neural networks through persistence of activation graphs](https://mlsanity.com/paper/2105.04404/) (2021)
- [Implicit Regularization of Accelerated Methods in Hilbert Spaces](https://mlsanity.com/paper/1905.13000/) (2019)
- [Learning and Inference in Hilbert Space with Quantum Graphical Models](https://mlsanity.com/paper/1810.12369/) (2018)
- [Continuous-time Value Function Approximation in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1806.02985/) (2018)

## Abstract (arXiv)

Various topological techniques and tools have been applied to neural networks in terms of network complexity, explainability, and performance. One fundamental assumption of this line of research is the existence of a global (Euclidean) coordinate system upon which the topological layer is constructed. Despite promising results, such a textittopologization method has yet to be widely adopted because the parametrization of a topologization layer takes a considerable amount of time and more importantly, lacks a theoretical foundation without which the performance of the neural network only achieves suboptimal performance. This paper proposes a learnable topological layer for neural networks without requiring a Euclidean space; Instead, the proposed construction requires nothing more than a general metric space except for an inner product, i.e., a Hilbert space. Accordingly, the according parametrization for the proposed topological layer is free of user-specified hyperparameters, which precludes the costly parametrization stage and the corresponding possibility of suboptimal networks.