Curvature as a tool for evaluating dimensionality reduction and estimating intrinsic dimension
This provides a novel geometric tool for assessing data representations and intrinsic dimensionality, with potential applications in network analysis and dimensionality reduction evaluation.
The authors introduced a curvature-based geometric profile for discrete metric spaces using abstract notions of sectional curvature, which captures metric relations between points. They demonstrated that this profile can quantitatively evaluate dimensionality reduction techniques and estimate the intrinsic dimensionality of datasets, applying it to analyze empirical networks.
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.