TopoFisher: Learning Topological Summary Statistics by Maximizing Fisher Information

2026-05-08

AI Summary

Traditional topological data analysis relies on handcrafted filtration and summarization steps, lacking an optimization objective directly tied to parameter uncertainty and thus struggling to efficiently extract information relevant for inference. This work proposes TopoFisher—the first differentiable persistent homology framework based on Fisher information maximization—which learns optimal topological representations in an end-to-end manner without requiring supervisory labels or posterior samples. The method jointly optimizes a trainable filtration function, a persistence diagram vectorization module, and a compression component, trained via gradient descent using a log-determinant Fisher loss. In high-dimensional non-Gaussian field inference tasks such as weak gravitational lensing, TopoFisher achieves performance comparable to unconstrained neural networks while using only 1/80th of the parameters and demonstrates enhanced robustness under simulator shifts.

Abstract

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 \textbf{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 $\sim80\times$ 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.

Problem

• simulation-based inference
• topological summary statistics
• Fisher information
• persistent homology
• parameter uncertainty

Innovation

• TopoFisher
• persistent homology
• Fisher information
• simulation-based inference
• topological data analysis