TopoFisher: Learning Topological Summary Statistics by Maximizing Fisher Information

2026-05-11 · Source: stat.ML updates on arXiv.org · Field: Science & Research — Mathematics & Computational Sciences, Space Science & Astronomy, Artificial Intelligence & Machine Learning · Depth: Expert, extended

Summary

TopoFisher is a novel differentiable persistent-homology pipeline designed to learn topological summary statistics by maximizing local Gaussian Fisher information. It optimizes trainable filtrations, diagram vectorizations, and compressors from simulations near a fiducial parameter value, without requiring posterior samples or supervised regression targets. The framework preserves the inductive bias of stable topological descriptors and provides regularity conditions for its log-determinant Fisher loss. Experiments on noisy spirals and Gaussian random fields demonstrate that TopoFisher recovers a significant fraction of available information, outperforming fixed topological vectorizations. In weak gravitational lensing, TopoFisher's learned topological summaries achieve a log-determinant Fisher score of approximately 21, surpassing power spectrum (13.8), peak counts (17.1), and wavelet scattering (19.3), and approaching an unconstrained Information Maximising Neural Network (IMNN) baseline (22.4) with up to 80 times fewer parameters. Notably, the fixed-filtration variant of TopoFisher exhibits superior generalization under simulator shifts and yields tighter constraints in neural posterior estimation compared to the neural baseline.

Key takeaway

For AI Scientists and Research Scientists working on simulation-based inference with complex, high-dimensional data, TopoFisher offers a robust and parameter-efficient alternative to black-box neural networks. You should consider integrating TopoFisher, particularly its fixed-filtration, learned-vectorization variant, into your pipelines to achieve strong in-distribution performance while significantly improving generalization under simulator shifts and yielding tighter posterior constraints. This approach can lead to more interpretable and reliable summary statistics for cosmological and other scientific inference tasks.

Key insights

TopoFisher optimizes topological summaries for simulation-based inference by maximizing Fisher information, offering robustness and parameter efficiency.

Principles

• Fisher information quantifies summary statistic utility.
• Topological inductive bias enhances generalization.
• Differentiable persistent homology enables gradient-based optimization.

Method

TopoFisher learns filtrations, vectorizations, and compressors by minimizing a negative log-determinant Gaussian Fisher loss, estimated from simulations near a fiducial parameter.

In practice

• Use TopoFisher for robust, parameter-efficient simulation-based inference.
• Prioritize learning diagram vectorization over complex filtrations for stability.
• Apply to high-dimensional, non-Gaussian data like weak gravitational lensing.

Topics

• TopoFisher
• Differentiable Persistent Homology
• Fisher Information Maximization
• Simulation-Based Inference
• Weak Gravitational Lensing

Code references

• RitAreaSciencePark/TopoFisher

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.