TopoFisher: Learning Topological Summary Statistics by Maximizing Fisher Information

Persistence diagrams provide stable, interpretable summaries of geometric and topological structure, and are useful for simulation-based inference when important information is not captured by low-order statistics. In practice, however, persistence-based pipelines require hand-chosen filtrations, vectorizations, and compressors, usually without an objective tied directly to parameter uncertainty. We introduce \textbf{TopoFisher}, a differentiable persistent-homology pipeline that learns topological summaries by maximizing local Gaussian Fisher information. From simulations near a fiducial parameter value, TopoFisher optimizes trainable filtrations, diagram vectorizations, and compressors without posterior samples or supervised regression targets, while preserving the inductive bias of stable topological descriptors. We also give sufficient regularity conditions under which the log-determinant Fisher loss is locally Lipschitz in the trainable parameters. Controlled experiments on noisy spirals and Gaussian random fields, where the total Fisher information is known, validate the pipeline: TopoFisher recovers a large fraction of the available information and improves over fixed topological vectorizations. Our main results are on weak gravitational lensing, a high-dimensional non-Gaussian field-inference problem from cosmology. There, both learned topological summaries, a fixed cubical filtration with a learned PersLay vectorization, and a learned CNN filtration with the same PersLay vectorization, reach $\log|F|\approx 21$, compared with $13.8$ for the power spectrum, $17.1$ for peak counts, and $19.3$ for wavelet scattering, approaching an unconstrained Information Maximising Neural Network baseline ($22.4$) with up to $\sim80\times$ fewer parameters. More importantly, the fixed-filtration variant generalizes better: under simulator shift from lognormal to LPT-based maps it retains $\log|F|=19.24$ while the neural baseline drops to $8.9$, and in neural posterior estimation it yields tighter constraints than the neural baseline, power spectrum, peak counts, and wavelet scattering. These results suggest Fisher-based topological optimizations as a robust, parameter-efficient front end for simulation-based inference.