17 citations · 26 across the 8 of their papers we have counts for
10 papers
Probabilistic Foundations of Fuzzy Simplicial Sets for Nonlinear Dimensionality Reduction
Janis Keck, Lukas Silvester Barth, Fatemeh +3
Fuzzy simplicial sets have become an object of interest in dimensionality reduction and manifold learning, most prominently through their role in UMAP. However, their definition th…
Curvature as a tool for evaluating dimensionality reduction and estimating intrinsic dimension
Charlotte Beylier, Parvaneh Joharinad, Jürgen Jost +1
Utilizing recently developed abstract notions of sectional curvature, we introduce a method for constructing a curvature-based geometric profile of discrete metric spaces. The curv…
Geometry and Dress groups with non-symmetric cost functions
Lukas Silvester Barth, Parvaneh Joharinad, Jürgen Jost +1
A metric relation by definition is symmetric. Since many data sets are non-symmetric, in this paper we develop a systematic theory of non-symmetric cost functions. Betweenness rela…
Merging Hazy Sets with m-Schemes: A Geometric Approach to Data Visualization
Lukas Silvester Barth, Hannaneh Fahimi, Parvaneh Joharinad +2
Many machine learning algorithms try to visualize high dimensional metric data in 2D in such a way that the essential geometric and topological features of the data are highlighted…
IsUMap: Manifold Learning and Data Visualization leveraging Vietoris-Rips filtrations
Lukas Silvester Barth, Fatemeh, Fahimi +3
This work introduces IsUMap, a novel manifold learning technique that enhances data representation by integrating aspects of UMAP and Isomap with Vietoris-Rips filtrations. We pres…
Fuzzy simplicial sets and their application to geometric data analysis
Lukas Silvester Barth, Hannaneh Fahimi, Parvaneh Joharinad +3
In this article, we expand upon the concepts introduced by David Spivak about the relationship between the category of uber metric spaces and the category $\mathbf{sF…
Reverse Bernstein Inequality on the Circle
Parvaneh Joharinad, Jürgen Jost, Sunhyuk Lim +1
The more then hundred years old Bernstein inequality states that the supremum norm of the derivative of a trigonometric polynomial of fixed degree can be bounded from above by supr…
Geometry of Data
Parvaneh Joharinad, Jürgen Jost
Topological data analysis asks when balls in a metric space intersect. Geometric data analysis asks how much balls have to be enlarged to intersect. We connect this princip…
Topological representation of the geometry of metric spaces
Parvaneh Joharinad, Jürgen Jost
Inspired by the concept of hyperconvexity and its relation to curvature, we translate geometric properties of a metric space encoded by the curvature inequalities into the persiste…
Topology and curvature of metric spaces
Parvaneh Joharinad, Jürgen Jost
We develop a new concept of non-positive curvature for metric spaces, based on intersection patterns of closed balls. In contrast to the synthetic approaches of Alexandrov and Bues…