用最优传输理论中的格罗莫夫-沃瑟斯坦距离做降维，更稳健地保留数据关系。
A dimensionality reduction technique based on the Gromov-Wasserstein distance
- 基于格罗莫夫-沃瑟斯坦距离构建新降维方法，从概率视角重新理解经典MDS和Isomap
- 通过梯度下降将高维数据嵌入低维空间，有效保留数据间结构关系
- 适合处理复杂高维数据，尤其对噪声敏感场景有更好鲁棒性
数据分析中，揭示对象间关系是核心问题。维度缩减（DR）技术用于生成更小、更易管理的数据表示。本文提出一种基于最优传输理论与格罗莫夫-沃瑟斯坦距离的新维度缩减方法。我们从概率角度重新阐释经典的多维标度（MDS）与非线性降维算法Isomap（等距映射），将高维数据的概率分布与其低维表示间的格罗莫夫-沃瑟斯坦距离作为优化目标。通过梯度下降，该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一种稳健且高效的新方案。
原文摘要 · Abstract (English)
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.
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