{"ID":2880797,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.14008","arxiv_id":"2508.14008","title":"Typed Topological Structures Of Datasets","abstract":"A datatset $X$ on $R^2$ is a finite topological space. Current research of a dataset focuses on statistical methods and the algebraic topological method \\cite{carlsson}. In \\cite{hu}, the concept of typed topological space was introduced and showed to have the potential for studying finite topological spaces, such as a dataset. It is a new method from the general topology perspective. A typed topological space is a topological space whose open sets are assigned types. Topological concepts and methods can be redefined using open sets of certain types. In this article, we develop a special set of types and its related typed topology on a dataset $X$. Using it, we can investigate the inner structure of $X$. In particular, $R^2$ has a natural quotient space, in which $X$ is organized into tracks, and each track is split into components. Those components are in a order. Further, they can be represented by an integer sequence. Components crossing tracks form branches, and the relationship can be well represented by a type of pseudotree (called typed-II pseudotree). Such structures provide a platform for new algorithms for problems such as calculating convex hull, holes, clustering and anomaly detection.","short_abstract":"A datatset $X$ on $R^2$ is a finite topological space. Current research of a dataset focuses on statistical methods and the algebraic topological method \\cite{carlsson}. In \\cite{hu}, the concept of typed topological space was introduced and showed to have the potential for studying finite topological spaces, such as a...","url_abs":"https://arxiv.org/abs/2508.14008","url_pdf":"https://arxiv.org/pdf/2508.14008v1","authors":"[\"Wanjun Hu\"]","published":"2025-08-19T17:14:13Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"math.GN\"]","methods":"[\"Generative Adversarial Network\"]","has_code":false}