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.

翻译：无监督机器学习缺乏定义上的真实标签。这为设计评估此类算法性能的指标带来了重大困难。与监督学习中已有大量质量度量指标研究的鲜明对比是，在降维领域仅提出了少数过度简化的指标。本研究旨在引入首个高度非平凡的降维性能度量指标。该指标基于黎曼几何中的截面曲率特性。为验证其可行性，该指标被用于评估当前最优的常用降维算法。此外，为使算法评估稳健且具有代表性，利用平面曲线的曲率性质，构建了一种以函数生成器形式呈现的新型参数化问题实例生成器。实验结果与基于被评估算法的设计特性及数据实例特征所预期的结果一致。