# "Normalized Stress" is Not Normalized: How to Interpret Stress Correctly

arXiv: https://arxiv.org/abs/2408.07724 · HTML: https://mlsanity.com/paper/2408.07724/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 74 references that are in the map.
- First posted on arXiv 14 Aug 2024.

## Builds on (most cited first, 11 of 11)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Interpretable Machine Learning: Fundamental Principles and 10 Grand Challenges](https://mlsanity.com/paper/2103.11251/) (2021)
- [Adversarial Autoencoders](https://mlsanity.com/paper/1511.05644/) (2015)
- [Visualizing Large-scale and High-dimensional Data](https://mlsanity.com/paper/1602.00370/) (2016)
- [A survey of dimensionality reduction techniques](https://mlsanity.com/paper/1403.2877/) (2014)
- [Measuring and Explaining the Inter-Cluster Reliability of Multidimensional Projections](https://mlsanity.com/paper/2107.07859/) (2021)
- [Multi-Perspective, Simultaneous Embedding](https://mlsanity.com/paper/1909.06485/) (2019)
- [ENS-t-SNE: Embedding Neighborhoods Simultaneously t-SNE](https://mlsanity.com/paper/2205.11720/) (2022)
- [nSimplex Zen: A Novel Dimensionality Reduction for Euclidean and Hilbert Spaces](https://mlsanity.com/paper/2302.11508/) (2023)
- [DiffRed: Dimensionality Reduction guided by stable rank](https://mlsanity.com/paper/2403.05882/) (2024)
- [Xtreaming: an incremental multidimensional projection technique and its application to streaming data](https://mlsanity.com/paper/2003.09017/) (2020)

## Built on by (most cited first, 5 of 5)

- [UMATO: Bridging Local and Global Structures for Reliable Visual Analytics with Dimensionality Reduction](https://mlsanity.com/paper/2508.16227/) (2025)
- [How Scale Breaks "Normalized Stress" and KL Divergence: Rethinking Quality Metrics](https://mlsanity.com/paper/2510.08660/) (2025)
- [Cluster and then Embed: A Modular Approach for Visualization](https://mlsanity.com/paper/2509.03373/) (2025)
- [Measuring Distortion in the Empty Regions of Dimensionality Reduction Scatterplots with the Gap Index](https://mlsanity.com/paper/2607.28324/) (2026)
- [Inductive Graph Layout with Implicit Neural Fields](https://mlsanity.com/paper/2608.08876/) (2026)

## Abstract (arXiv)

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.