Signatures, Lipschitz-Free Spaces, and Paths of Persistence Diagrams
Bibliographic record
Abstract
Abstract. Paths of persistence diagrams provide a summary of the dynamic topological structure of a one-parameter family of metric spaces. These summaries can be used to study and characterize the dynamic shape of data such as swarming behavior in multiagent systems, time-varying functional MRI scans from neuroscience, and time-dependent scalar fields in hydrodynamics. While persistence diagrams can provide a powerful topological summary of data, the standard space of persistence diagrams lacks the sufficient algebraic and analytic structure required for many theoretical and computational analyses. We enrich the space of persistence diagrams by isometrically embedding it into a Lipschitz-free space, a Banach space built from a universal construction. We utilize the Banach space structure to define bounded variation paths of persistence diagrams, which can be studied using the path signature, a reparametrization-invariant characterization of paths valued in a Banach space. The signature is universal and characteristic, which allow us to theoretically characterize measures on the space of paths and motivate its use in the context of kernel methods. However, kernel methods often require a feature map into a Hilbert space, so we introduce the moment map, a stable and injective feature map for static persistence diagrams, and compose it with the discrete path signature, producing a computable feature map into a Hilbert space. Finally, we demonstrate the efficacy of our methods by applying this to a parameter estimation problem for a 3-dimensional model of swarming behavior.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".