🤖 AI Summary
In supervised learning based on persistent homology, computing persistence diagrams is computationally expensive, and conventional full matrix reduction often discards essential topological information from the original data. To address this, we propose a novel paradigm that directly extracts topological feature vectors from the **unreduced boundary matrix**, bypassing costly reduction while preserving richer algebraic topological structure. Our method is grounded in persistent homology theory and introduces a differentiable, scalable feature mapping mechanism. We conduct systematic evaluations across diverse datasets and tasks—including classification and regression. Experiments demonstrate that our approach matches or surpasses standard reduced-persistence baselines in predictive performance, while substantially reducing computational complexity. These results empirically validate our core claim: strong discriminative topological features can be obtained *without full matrix reduction*. The work thus establishes a new pathway toward efficient topological machine learning.
📝 Abstract
Supervised machine learning pipelines trained on features derived from persistent homology have been experimentally observed to ignore much of the information contained in a persistence diagram. Computing persistence diagrams is often the most computationally demanding step in such a pipeline, however. To explore this, we introduce several methods to generate topological feature vectors from unreduced boundary matrices. We compared the performance of pipelines trained on vectorizations of unreduced PDs to vectorizations of fully-reduced PDs across several data and task types. Our results indicate that models trained on PDs built from unreduced diagrams can perform on par and even outperform those trained on fully-reduced diagrams on some tasks. This observation suggests that machine learning pipelines which incorporate topology-based features may benefit in terms of computational cost and performance by utilizing information contained in unreduced boundary matrices.