Analyzing relationships between objects is a pivotal problem within data science. In this context, Dimensionality reduction (DR) techniques are employed to generate smaller and more manageable data representations. This paper proposes a new method for dimensionality reduction, based on optimal transportation theory and the Gromov-Wasserstein distance. We offer a new probabilistic view of the classical Multidimensional Scaling (MDS) algorithm and the nonlinear dimensionality reduction algorithm, Isomap (Isometric Mapping or Isometric Feature Mapping) that extends the classical MDS, in which we use the Gromov-Wasserstein distance between the probability measure of high-dimensional data, and its low-dimensional representation. Through gradient descent, our method embeds high-dimensional data into a lower-dimensional space, providing a robust and efficient solution for analyzing complex high-dimensional datasets.

翻译：分析对象间关系是数据科学中的关键问题。在此背景下，降维技术被用于生成更小且更易处理的数据表示。本文基于最优传输理论和Gromov-Wasserstein距离，提出了一种新的降维方法。我们为经典多维尺度分析算法及其非线性扩展算法——等距映射提供了新的概率视角，其中我们使用了高维数据概率测度与其低维表示之间的Gromov-Wasserstein距离。通过梯度下降法，我们的方法将高维数据嵌入低维空间，为分析复杂高维数据集提供了鲁棒且高效的解决方案。