Manifold Learning Projection Quality Quantitative Evaluation
Vladislav Belov, Radek Mařík · 2021
A large number dimensions may cause a variety of problems in real-world applications: some dimensions might be redundant and can worsen the quality of the workflow output, and, in the vast majority of exercises with datasets, data are distributed along a highly nonlinear manifold whose structure is unknown. This paper focuses on analyzing the outputs of nonlinear dimensionality reduction, or Manifold Learning, techniques. We introduce three meaningful measures that are capable of providing context behind projections onto lower-dimensional spaces. The measures will enable us to compare techniques with each other and assist in choosing suitable hyperparameters. Moreover, we propose to view projections from the standpoint of simplicial complex distortion. In connection to that, we establish the process of a dimension-agnostic graph-based data tessellation technique that builds a simplicial skeleton of high-dimensional data. Alongside our new tessellation technique, we evaluate the proposed quality measures on the Delaunay-tessellation-based simplicial approximations of manifolds.