Persistence codebooks for topological data analysis

Prediction of nanocomposite properties and process optimization using persistent homology and machine learning

Physical property prediction and synthesis process optimization are key targets in material informatics. In this study, we propose a machine learning approach that utilizes ridge regression to predict the oxygen permeability at fuel cell electrode surfaces and determine the optimal process temperature. These predictions are based on a persistence diagram derived from tomographic images captured using transmission electron microscopy (TEM). Through machine learning analysis of the complex structures present in the Pt/CeO2 nanocomposites, we discovered that l2 regularization considering diverse structural elements is more appropriate than l1 regularization (sparse modeling). Notably, our model successfully captured the activation energy of oxygen permeability, a phenomenon that could not be solely explained by the geometric feature of the Betti numbers, as demonstrated in a previous study. The correspondence between the ridge regression coefficient and persistence diagram revealed the formation process of the local and three-dimensional structures of CeO2 and their contributions to pre-exponential factor and activation energies. This analysis facilitated the determination of the annealing temperature required to achieve the optimal structure and accurately predict the physical properties.

A Survey of Vectorization Methods in Topological Data Analysis

Attempts to incorporate topological information in supervised learning tasks have resulted in the creation of several techniques for vectorizing persistent homology barcodes. In this paper, we study thirteen such methods. Besides describing an organizational framework for these methods, we comprehensively benchmark them against three well-known classification tasks. Surprisingly, we discover that the best-performing method is a simple vectorization, which consists only of a few elementary summary statistics. Finally, we provide a convenient web application which has been designed to facilitate exploration and experimentation with various vectorization methods.

ASPECTS OF TOPOLOGICAL APPROACHES FOR DATA SCIENCE

We establish a new theory which unifies various aspects of topological approaches for data science, by being applicable both to point cloud data and to graph data, including networks beyond pairwise interactions. We generalize simplicial complexes and hypergraphs to super-hypergraphs and establish super-hypergraph homology as an extension of simplicial homology. Driven by applications, we also introduce super-persistent homology.