# How Scale Breaks "Normalized Stress" and KL Divergence: Rethinking Quality Metrics

arXiv: https://arxiv.org/abs/2510.08660 · HTML: https://mlsanity.com/paper/2510.08660/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 86 references that are in the map.
- First posted on arXiv 9 Oct 2025.

## Builds on (most cited first, 19 of 19)

- [You Only Look Once: Unified, Real-Time Object Detection](https://mlsanity.com/paper/1506.02640/) (2015)
- [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)
- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [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)
- [Unveiling High-dimensional Backstage: A Survey for Reliable Visual Analytics with Dimensionality Reduction](https://mlsanity.com/paper/2501.10168/) (2025)
- [ZADU: A Python Library for Evaluating the Reliability of Dimensionality Reduction Embeddings](https://mlsanity.com/paper/2308.00282/) (2023)
- [Classes are not Clusters: Improving Label-based Evaluation of Dimensionality Reduction](https://mlsanity.com/paper/2308.00278/) (2023)
- [SmartGD: A GAN-Based Graph Drawing Framework for Diverse Aesthetic Goals](https://mlsanity.com/paper/2206.06434/) (2022)
- [Multi-Perspective, Simultaneous Embedding](https://mlsanity.com/paper/1909.06485/) (2019)
- ["Normalized Stress" is Not Normalized: How to Interpret Stress Correctly](https://mlsanity.com/paper/2408.07724/) (2024)
- [ShaRP: Shape-Regularized Multidimensional Projections](https://mlsanity.com/paper/2306.00554/) (2023)
- [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, 1 of 1)

- [Class Angular Distortion Index for Dimensionality Reduction](https://mlsanity.com/paper/2605.00637/) (2026)

## Abstract (arXiv)

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.