An evaluation framework for dimensionality reduction through sectional curvature
This work addresses a fundamental problem in unsupervised machine learning by providing a more sophisticated evaluation framework for dimensionality reduction, though it appears incremental as it builds on existing geometric concepts without claiming broad SOTA breakthroughs.
The authors tackled the lack of robust evaluation metrics for unsupervised dimensionality reduction by introducing a novel metric based on sectional curvature from Riemannian geometry, and they tested it on common algorithms with a new parameterized problem instance generator, yielding experimental results consistent with algorithm designs and data features.
Unsupervised machine learning lacks ground truth by definition. This poses a major difficulty when designing metrics to evaluate the performance of such algorithms. In sharp contrast with supervised learning, for which plenty of quality metrics have been studied in the literature, in the field of dimensionality reduction only a few over-simplistic metrics has been proposed. In this work, we aim to introduce the first highly non-trivial dimensionality reduction performance metric. This metric is based on the sectional curvature behaviour arising from Riemannian geometry. To test its feasibility, this metric has been used to evaluate the performance of the most commonly used dimension reduction algorithms in the state of the art. Furthermore, to make the evaluation of the algorithms robust and representative, using curvature properties of planar curves, a new parameterized problem instance generator has been constructed in the form of a function generator. Experimental results are consistent with what could be expected based on the design and characteristics of the evaluated algorithms and the features of the data instances used to feed the method.