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1-parameter persistent homology, a cornerstone in Topological Data Analysis (TDA), studies the evolution of topological features such as connected components and cycles hidden in data. It has been applied to enhance the representation power of deep learning models, such as Graph Neural Networks (GNNs). To enrich the representations of topological features, here we propose to study 2-parameter persistence modules induced by bi-filtration functions. In order to incorporate these representations into machine learning models, we introduce a novel vector representation called Generalized Rank Invariant Landscape (GRIL) for 2-parameter persistence modules. We show that this vector representation is 1-Lipschitz stable and differentiable with respect to underlying filtration functions and can be easily integrated into machine learning models to augment encoding topological features. We present an algorithm to compute the vector representation efficiently. We also test our methods on synthetic and benchmark graph datasets, and compare the results with previous vector representations of 1-parameter and 2-parameter persistence modules. Further, we augment GNNs with GRIL features and observe an increase in performance indicating that GRIL can capture additional features enriching GNNs. We make the complete code for the proposed method available at https://github.com/soham0209/mpml-graph.
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Persistence Images: A Stable Vector Representation of Persistent Homology
Many data sets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a multiscale description of the homological features within a data set. A useful representation of this homological information is a persistence diagram (PD). Efforts have been made to map PDs into spaces with additional structure valuable to machine learning tasks. We convert a PD to a finite dimensional vector representation which we call a persistence image (PI), and prove the stability of this transformation with respect to small perturbations in the inputs. The discriminatory power of PIs is compared against existing methods, showing significant performance gains. We explore the use of PIs with vector-based machine learning tools, such as linear sparse support vector machines, which identify features containing discriminating topological information. Finally, high accuracy inference of parameter values from the dynamic output of a discrete dynamical system (the linked twist map) and a partial differential equation (the anisotropic Kuramoto-Sivashinsky equation) provide a novel application of the discriminatory power of PIs.
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- Award ID(s):
- 1633830
- PAR ID:
- 10045396
- Date Published:
- Journal Name:
- Journal of machine learning research
- Volume:
- 18
- ISSN:
- 1533-7928
- Page Range / eLocation ID:
- 1-35
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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