Complex, high-dimensional data is ubiquitous across many scientific disciplines, including machine learning, biology, and the social sciences. One of the primary methods of visualizing these datasets is with two-dimensional scatter plots that visually capture some properties of the data. Because visually determining the accuracy of these plots is challenging, researchers often use quality metrics to measure the projection's accuracy and faithfulness to the original data. One of the most commonly employed metrics, normalized stress, is sensitive to uniform scaling (stretching, shrinking) of the projection, despite this act not meaningfully changing anything about the projection. Another quality metric, the Kullback--Leibler (KL) divergence used in the popular t-Distributed Stochastic Neighbor Embedding (t-SNE) technique, is also susceptible to this scale sensitivity. We investigate the effect of scaling on stress and KL divergence analytically and empirically by showing just how much the values change and how this affects dimension reduction technique evaluations. We introduce a simple technique to make both metrics scale-invariant and show that it accurately captures expected behavior on a small benchmark.

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.

Stress is among the most commonly employed quality metrics and optimization criteria for dimension reduction projections of high dimensional data. Complex, high dimensional data is ubiquitous across many scientific disciplines, including machine learning, biology, and the social sciences. One of the primary methods of visualizing these datasets is with two dimensional scatter plots that visually capture some properties of the data. Because visually determining the accuracy of these plots is challenging, researchers often use quality metrics to measure projection accuracy or faithfulness to the full data. One of the most commonly employed metrics, normalized stress, is sensitive to uniform scaling of the projection, despite this act not meaningfully changing anything about the projection. We investigate the effect of scaling on stress and other distance based quality metrics analytically and empirically by showing just how much the values change and how this affects dimension reduction technique evaluations. We introduce a simple technique to make normalized stress scale invariant and show that it accurately captures expected behavior on a small benchmark.