“Normalized Stress” is Not Normalized: How to Interpret Stress Correctly

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 the projection's accuracy or faithfulness to the full 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. 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.