Topological Machine Learning with Persistence Indicator Functions

Rieck, Bastian; Sadlo, Filip; Leitte, Heike

doi:10.1007/978-3-030-43036-8_6

Cited by 28 publications

(24 citation statements)

References 16 publications

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“…More precisely, the Betti curve of dimension q of a diagram Q refers to the sequence of Betti numbers, i.e., active topological features, of dimension q in Q , evaluated for each threshold ϵ . It is a useful descriptor for numerous machine learning tasks[34]. Intuitively, the Betti numbers represent the number of q -dimensional holes in a topological space.…”

Section: Resultsmentioning

confidence: 99%

G2 stem cells orchestrate time-directed, long-range coordination of calcium signaling during skin epidermal regeneration

Moore

Gao

Matte-Martone

et al. 2021

Preprint

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Regenerative processes in the mammalian skin require coordinated cell-cell communication. Ca2+ signaling can coordinate tissue-level responses in developing and wounded epithelia in tissue explants and invertebrates. However, its role in the homeostatic, regenerative basal layer of the skin epithelium is unknown due to significant challenges in studying signaling dynamics in a spatially complex tissue context in live mice. Here we combine in vivo imaging of dynamic Ca2+ signaling at the single cell level across thousands of cells with a novel computational approach, Geometric Scattering Trajectory Homology (GSTH). GSTH models Ca2+ as signals over a cell adjacency graph and uses a multi-level wavelet-like transform (called a scattering transform) to extract signaling patterns from our high dimensional in vivo datasets. We discover local Ca2+ signaling patterns are orchestrated so that signals flow in a coordinated and directed manner across the tissue, distinct from topographically uncoordinated Ca2+ signaling in excitatory tissues. Directed Ca2+ signaling is regulated by the major gap junction protein in the epidermal stem cell layer, Connexin 43 (Cx43). Cx43 gap junctions are dissociated as cells progress through the cell cycle out of G1 and play an essential role in the progression of stem cells from G2 towards mitosis. Finally, G2 cells display related signaling patterns and are essential for tissue-level signaling coordination. Together, our results provide insight into how such a ubiquitous signaling pathway regulates highly specific behaviors and outcomes at a tissue-wide level to maintain proper homeostasis.

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Section: Resultsmentioning

confidence: 99%

G2 stem cells orchestrate time-directed, long-range coordination of calcium signaling during skin epidermal regeneration

Moore

Gao

Matte-Martone

et al. 2021

Preprint

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“…is the indicator function. The Betti curve was often informally used to analyse data (Umeda, 2017 ); recently, Rieck et al ( 2020a ) provided a summarising description of their features. Figure 5 depicts a simple illustration of the calculation of Betti curves.…”

Section: Surveymentioning

confidence: 99%

A Survey of Topological Machine Learning Methods

Hensel

Moor

Rieck

2021

Front. Artif. Intell.

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112

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The last decade saw an enormous boost in the field of computational topology: methods and concepts from algebraic and differential topology, formerly confined to the realm of pure mathematics, have demonstrated their utility in numerous areas such as computational biology personalised medicine, and time-dependent data analysis, to name a few. The newly-emerging domain comprising topology-based techniques is often referred to as topological data analysis (TDA). Next to their applications in the aforementioned areas, TDA methods have also proven to be effective in supporting, enhancing, and augmenting both classical machine learning and deep learning models. In this paper, we review the state of the art of a nascent field we refer to as “topological machine learning,” i.e., the successful symbiosis of topology-based methods and machine learning algorithms, such as deep neural networks. We identify common threads, current applications, and future challenges.

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“…Rieck et al . [RSL20b] defined a family of distances for Betti curves (also called the persistence indicator functions ), as well as corresponding kernels in order to use Betti curve in machine learning algorithms. Zhao and Wang [ZW19] introduced a weighted‐kernel for persistence images (WKPI), its induced distance, and a metric‐learning framework to learn the weights (and kernel) from labeled data.…”

Section: Comparative Measures For Topological Descriptorsmentioning

confidence: 99%

“…The persistent homology transform (PHT) introduced by Turner et al [TMB14] comes with a distance measure, referred to as the PHT distance, which captures similarity between shapes in shape classification. The inter-level set persistence hierarchies (ISPHs) [RSL17,RSL20b] are directed trees, whose similarity can be measured by the edit distance (see Sect. 4.2).…”

Section: Comparing Persistence Diagrams and Their Variantsmentioning

confidence: 99%

Scalar Field Comparison with Topological Descriptors: Properties and Applications for Scientific Visualization

Yan

Masood

Sridharamurthy

et al. 2021

Computer Graphics Forum

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In topological data analysis and visualization, topological descriptors such as persistence diagrams, merge trees, contour trees, Reeb graphs, and Morse–Smale complexes play an essential role in capturing the shape of scalar field data. We present a state‐of‐the‐art report on scalar field comparison using topological descriptors. We provide a taxonomy of existing approaches based on visualization tasks associated with three categories of data: single fields, time‐varying fields, and ensembles. These tasks include symmetry detection, periodicity detection, key event/feature detection, feature tracking, clustering, and structure statistics. Our main contributions include the formulation of a set of desirable mathematical and computational properties of comparative measures, and the classification of visualization tasks and applications that are enabled by these measures.

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Topological Machine Learning with Persistence Indicator Functions

Cited by 28 publications

References 16 publications

G2 stem cells orchestrate time-directed, long-range coordination of calcium signaling during skin epidermal regeneration

G2 stem cells orchestrate time-directed, long-range coordination of calcium signaling during skin epidermal regeneration

A Survey of Topological Machine Learning Methods

Scalar Field Comparison with Topological Descriptors: Properties and Applications for Scientific Visualization

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