Hierarchical Clustering in $Λ$CDM Cosmologies via Persistence Energy

In this research, we investigate the structural evolution of the cosmic web, employing advanced methodologies from Topological Data Analysis. Our approach involves leveraging LITE, an innovative method from recent literature that embeds persistence diagrams into elements of vector spaces. Utilizing this methodology, we analyze three quintessential cosmic structures: clusters, filaments, and voids. A central discovery is the correlation between \textit{Persistence Energy} and redshift values, linking persistent homology with cosmic evolution and providing insights into the dynamics of cosmic structures.

LITE: A Stable Framework for Lattice-Integrated Embedding of Topological Descriptors

In this paper, we introduce a new family of descriptors for persistence diagrams. Our approach transforms these diagrams into elements of a finite-dimensional vector space using functionals based on the discrete measures they induce. While our focus is primarily on identity and frequency-based transformations, we do not restrict our approach exclusively to this types of techniques. We term this family of transformations as LITE (Lattice Integrated Topological Embedding) and prove stability for some members of this family against the 1-$Kantorovitch$-$Rubinstein$ metric, ensuring its responsiveness to subtle data variations. Extensive comparative analysis reveals that our descriptor performs competitively with the current state-of-art from the topological data analysis literature, and often surpasses, the existing methods. This research not only introduces an innovative perspective for data scientists but also critiques the current trajectory of literature on methodologies for vectorizing diagrams. It establishes a foundation for future progress in applying persistence diagrams to data analysis and machine learning under a more simple and effective lens.