On some variations of TC and the LS-category

Abstract: Datasets can be viewed as mathematical objects (e.g., point clouds, matrices, graphs, images, fields/functions) that have shape. This shape can describe the space that data populates (e.g., data that lies on a manifold) or can be used to understand the complex structures contained within data (e.g., the multi-scale organization of self-assembled materials). Data shape can be exploited to improve the effectiveness of data analysis methods or provide connections between complex materials and their physical and chemical properties. However, quantifying shape is difficult to do with common methods based on statistics, signal processing, or with the use of machine learning. Topology and geometry are fields of mathematics that provide tools for the characterization and quantification of the shape of data directly.

In this talk I will discuss how data taken from industrial processes, such as time series and images, can be represented as a shape and how that shape can be analyzed through topological and geometrical methods such as the Euler characteristic (EC) and Riemannian manifold geometry. I will provide a brief overview of these methods and illustrate how exploiting the topology and geometry of data can provide improvements in data-centric tasks such as dimensionality reduction, anomaly detection, and statistical process control in the context of textile production, chemical process systems, and granular material manufacturing.

geometric topology

Audience: researchers in the topic

( video )

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GEOTOP-A seminar

Series comments: Web-seminar series on Applications of Geometry and Topology

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