Barbara Giunti

Martin Uray, Barbara Giunti, Michael Kerber et al.

Topological Data Analysis (TDA) is a discipline that applies algebraic topology techniques to analyze complex, multi-dimensional data. Although it is a relatively new field, TDA has been widely and successfully applied across various domains, such as medicine, materials science, and biology. This survey provides an overview of the state of the art of TDA within a dynamic and promising application area: industrial manufacturing and production, particularly within the Industry 4.0 context. We have conducted a rigorous and reproducible literature search focusing on TDA applications in industrial production and manufacturing settings. The identified works are categorized based on their application areas within the manufacturing process and the types of input data. We highlight the principal advantages of TDA tools in this context, address the challenges encountered and the future potential of the field. Furthermore, we identify TDA methods that are currently underexploited in specific industrial areas and discuss how their application could be beneficial, with the aim of stimulating further research in this field. This work seeks to bridge the theoretical advancements in TDA with the practical needs of industrial production. Our goal is to serve as a guide for practitioners and researchers applying TDA in industrial production and manufacturing systems. We advocate for the untapped potential of TDA in this domain and encourage continued exploration and research.

Barbara Giunti, Jānis Lazovskis

The barcode of a filtration and its representative cycles encode rich information often useful in data analysis. However, obtaining them can be computationally expensive. Therefore, it is useful to have methods that update them if the associated filtration undergoes small changes. There are already efficient algorithms updating a barcode if simplices exchange entrance order or are added, but not if simplices are removed. We provide an implementation to update a reduced boundary matrix when simplices in the filtration are removed. Our algorithm, the Simplicial Removal Update Procedure (SiRUP), intrinsically updates also the representative cycles, and is compatible with the clearing optimizations. We show that the complexity of our algorithm is lower than recomputing the barcode from scratch and that the number of executed matrix column additions is minimal, with both theoretical and experimental methods.

Nithisha Raghavaraju, Barbara Giunti, Bastian Rieck

Chatbots have become increasingly prevalent across various domains, offering automated assistance in many areas, especially mental health support. The training is done using extremely large datasets, which are sometimes not available in very specific domains. Moreover, it would sometimes be ideal to train the chatbot with personal information about the patients, which, of course, cannot be done on shared servers since it would violate patient confidentiality. Hence, being able to improve the performance of a chatbot, possibly trained locally and on a restricted dataset, without having to increase the dataset itself, would be extremely beneficial. In this work, we will enhance the input datasets using persistent homology (PH) vectorizations computed from the raw datasets themselves. Then we will compare, across several metrics, the performance of multiple chatbot models with or without the PH enhancement. Our experiments suggest that, while at times the PH enhancement is not particularly beneficial, it sometimes brings remarkable advantages for virtually no cost.

Barbara Giunti, Elizabeth Munch

In this paper, we provide a new construction for studying parameterized persistence, called a canopy. We give two versions of this construction: the A-canopy, retaining all information about points on the diagonal of the persistence diagram; and the D-canopy, encoding the information of the "standard" persistence diagram. We do this by making a simple but major modification in the persistence bundle representation information: namely, rather than tracking a point in the persistence diagram, we instead track some choice of pairs of simplices that created said point. This viewpoint is a combinatorial version of tracking the chain complex information rather than just the output of persistence. We show how to construct the canopies from any filtered filtration function, proving, using the algebraic structure of filtered chain complexes, that different choices of pairs result in homeomorphic structures. Finally, we showcase the power of our approach by using canopies to define vines even in the presence of points with multiplicity; to discuss monodromy; and to obtain some immediate results linking non-trivial monodromy in the persistent homology transform with the existence of non-Hausdorff points in the canopy.