Curvature as a tool for evaluating dimensionality reduction and estimating intrinsic dimension

• URL: http://arxiv.org/abs/2509.13385v1
• Date: Tue, 16 Sep 2025 10:14:47 GMT
• Title: Curvature as a tool for evaluating dimensionality reduction and estimating intrinsic dimension
• Authors: Charlotte Beylier, Parvaneh Joharinad, Jürgen Jost, Nahid Torbati,
• Abstract summary: We introduce a method for constructing a curvature-based geometric profile of discrete metric spaces.<n>More significantly, based on this curvature profile, we introduce a quantitative measure to evaluate the effectiveness of data representations.<n>Our experiments demonstrate that this curvature-based analysis can be employed to estimate the intrinsic dimensionality of datasets.
• Score: 1.3019517863608956
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Utilizing recently developed abstract notions of sectional curvature, we introduce a method for constructing a curvature-based geometric profile of discrete metric spaces. The curvature concept that we use here captures the metric relations between triples of points and other points. More significantly, based on this curvature profile, we introduce a quantitative measure to evaluate the effectiveness of data representations, such as those produced by dimensionality reduction techniques. Furthermore, Our experiments demonstrate that this curvature-based analysis can be employed to estimate the intrinsic dimensionality of datasets. We use this to explore the large-scale geometry of empirical networks and to evaluate the effectiveness of dimensionality reduction techniques.

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