# TopoFisher: Learning Topological Summary Statistics by Maximizing Fisher Information

arXiv: https://arxiv.org/abs/2605.07720 · HTML: https://mlsanity.com/paper/2605.07720/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 55 references that are in the map.
- First posted on arXiv 8 May 2026.

## Builds on (most cited first, 17 of 17)

- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [Automatic Posterior Transformation for Likelihood-Free Inference](https://mlsanity.com/paper/1905.07488/) (2019)
- [Flexible statistical inference for mechanistic models of neural dynamics](https://mlsanity.com/paper/1711.01861/) (2017)
- [PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures](https://mlsanity.com/paper/1904.09378/) (2019)
- [Deep Learning with Topological Signatures](https://mlsanity.com/paper/1707.04041/) (2017)
- [A Topology Layer for Machine Learning](https://mlsanity.com/paper/1905.12200/) (2019)
- [Sliced Wasserstein Kernel for Persistence Diagrams](https://mlsanity.com/paper/1706.03358/) (2017)
- [Graph Filtration Learning](https://mlsanity.com/paper/1905.10996/) (2019)
- [Kymatio: Scattering Transforms in Python](https://mlsanity.com/paper/1812.11214/) (2018)
- [Stochastic Convergence of Persistence Landscapes and Silhouettes](https://mlsanity.com/paper/1312.0308/) (2013)
- [Persistence Fisher Kernel: A Riemannian Manifold Kernel for Persistence Diagrams](https://mlsanity.com/paper/1802.03569/) (2018)
- [A Framework for Differential Calculus on Persistence Barcodes](https://mlsanity.com/paper/1910.00960/) (2019)
- [Topologically penalized regression on manifolds](https://mlsanity.com/paper/2110.13749/) (2021)
- [Scattering Transform Based Image Clustering using Projection onto Orthogonal Complement](https://mlsanity.com/paper/2011.11586/) (2020)
- [Bayesian Simulation-based Inference for Cosmological Initial Conditions](https://mlsanity.com/paper/2310.19910/) (2023)

## Abstract (arXiv)

Persistence diagrams provide stable, interpretable summaries of geometric and topological structure and are useful for simulation-based inference when low-order statistics miss key information. Yet persistence-based pipelines require hand-chosen filtrations, vectorizations, and compressors, typically without an objective tied to parameter uncertainty. We introduce TopoFisher, a differentiable persistent-homology pipeline that learns topological summaries by maximizing local Gaussian Fisher information. Using simulations near a fiducial parameter, TopoFisher optimizes trainable filtrations, diagram vectorizations, and compressors without posterior samples or supervised regression targets, while retaining stable topological inductive bias. We also give sufficient regularity conditions for the log-determinant Fisher loss to be locally Lipschitz in trainable parameters. Controlled experiments on noisy spirals and Gaussian random fields, where total Fisher information is known, show that TopoFisher recovers much of the available information and outperforms fixed topological vectorizations. Our main results are on weak gravitational lensing, a high-dimensional non-Gaussian cosmological field-inference problem. Learned topological summaries reach higher Fisher information than state-of-the-art cosmological summaries and approach an unconstrained Information Maximising Neural Network baseline with up to ~80× fewer parameters. The learned filtrations also generalize better: under simulator shift from lognormal to LPT-based maps it retains most Fisher information, while the neural baseline drops, and in neural posterior estimation they give tighter constraints than the neural baseline, and of state-of-the-art cosmological summaries. These results support Fisher-based topological optimization as a robust, parameter-efficient front end for simulation-based inference.