标题： 拓扑：具有抗非度量性和稀疏性鲁棒性的差异数据的力导向欧几里得嵌入 显示英文标题

标题： Topolow: Force-Directed Euclidean Embedding of Dissimilarity Data with Robustness Against Non-Metricity and Sparsity

摘要： 基于成对差异矩阵将一组对象嵌入到低维欧几里得空间中的问题是数据分析、机器学习和统计学中的基本问题。 然而，当输入的差异不满足度量或欧几里得公理时，许多标准分析方法的假设会被违反。 我们提出了Topolow的数学和统计基础，Topolow是一种受物理学启发的、无需梯度的优化框架，用于此类嵌入问题。 Topolow在概念上与力导向图绘制算法相关，但其目标是定量度量重建，这使其根本区别于其他方法。 它将对象建模为物理系统中的粒子，并通过顺序的、随机的成对交互进行新颖的优化方案，从而避免计算全局梯度，并在面对稀疏数据时对收敛到局部最优具有鲁棒性。 Topolow在拉普拉斯误差模型下最大化似然，对异常值和异质误差具有鲁棒性，并能正确处理截断数据。 至关重要的是，Topolow不需要输入的差异是度量的，这使其成为将非度量测量嵌入有效欧几里得空间的鲁棒解决方案，从而使得可以使用标准分析工具。 我们展示了Topolow在重建稀疏和非欧几里得数据几何结构方面相比标准多维尺度分析（MDS）方法的优越性能。 本文正式化了该算法，该算法最初在抗原映射背景下以Topolow的形式提出（Arhami和Rohani，2025）（开放获取），并着重介绍了其度量嵌入和数学特性，以供更广泛的受众了解。 通用函数Euclidify可在R包topolow中使用。 显示英文摘要

摘要： The problem of embedding a set of objects into a low-dimensional Euclidean space based on a matrix of pairwise dissimilarities is fundamental in data analysis, machine learning, and statistics. However, the assumptions of many standard analytical methods are violated when the input dissimilarities fail to satisfy metric or Euclidean axioms. We present the mathematical and statistical foundations of Topolow, a physics-inspired, gradient-free optimization framework for such embedding problems. Topolow is conceptually related to force-directed graph drawing algorithms but is fundamentally distinguished by its goal of quantitative metric reconstruction. It models objects as particles in a physical system, and its novel optimization scheme proceeds through sequential, stochastic pairwise interactions, which circumvents the need to compute a global gradient and provides robustness against convergence to local optima, especially for sparse data. Topolow maximizes the likelihood under a Laplace error model, robust to outliers and heterogeneous errors, and properly handles censored data. Crucially, Topolow does not require the input dissimilarities to be metric, making it a robust solution for embedding non-metric measurements into a valid Euclidean space, thereby enabling the use of standard analytical tools. We demonstrate the superior performance of Topolow compared to standard Multidimensional Scaling (MDS) methods in reconstructing the geometry of sparse and non-Euclidean data. This paper formalizes the algorithm, first introduced as Topolow in the context of antigenic mapping in (Arhami and Rohani, 2025) (open access), with emphasis on its metric embedding and mathematical properties for a broader audience. The general-purpose function Euclidify is available in the R package topolow.