# The magnitude vector of images

arXiv: https://arxiv.org/abs/2110.15188 · HTML: https://mlsanity.com/paper/2110.15188/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 39 references that are in the map.
- First posted on arXiv 28 Oct 2021.

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

- [ChestX-ray8: Hospital-scale Chest X-ray Database and Benchmarks on Weakly-Supervised Classification and Localization of Common Thorax Diseases](https://mlsanity.com/paper/1705.02315/) (2017)
- [Feature Squeezing: Detecting Adversarial Examples in Deep Neural Networks](https://mlsanity.com/paper/1704.01155/) (2017)
- [Deep metric learning using Triplet network](https://mlsanity.com/paper/1412.6622/) (2014)
- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [Deep Metric Learning with Hierarchical Triplet Loss](https://mlsanity.com/paper/1810.06951/) (2018)
- [Dense Extreme Inception Network: Towards a Robust CNN Model for Edge Detection](https://mlsanity.com/paper/1909.01955/) (2019)
- [A Refined Laser Method and Faster Matrix Multiplication](https://mlsanity.com/paper/2010.05846/) (2020)
- [Unmixing Convolutional Features for Crisp Edge Detection](https://mlsanity.com/paper/2011.09808/) (2020)
- [Uncovering the Topology of Time-Varying fMRI Data using Cubical Persistence](https://mlsanity.com/paper/2006.07882/) (2020)
- [Topological Machine Learning with Persistence Indicator Functions](https://mlsanity.com/paper/1907.13496/) (2019)
- [Weighting vectors for machine learning: numerical harmonic analysis applied to boundary detection](https://mlsanity.com/paper/2106.00827/) (2021)

## Built on by (most cited first, 4 of 4)

- [Metric Space Magnitude for Evaluating the Diversity of Latent Representations](https://mlsanity.com/paper/2311.16054/) (2023)
- [Metric Space Magnitude and Generalisation in Neural Networks](https://mlsanity.com/paper/2305.05611/) (2023)
- [The magnitude of categories of texts enriched by language models](https://mlsanity.com/paper/2501.06662/) (2025)
- [Geometry-Aware Edge Pooling for Graph Neural Networks](https://mlsanity.com/paper/2506.11700/) (2025)

## Abstract (arXiv)

The magnitude of a finite metric space has recently emerged as a novel invariant quantity, allowing to measure the effective size of a metric space. Despite encouraging first results demonstrating the descriptive abilities of the magnitude, such as being able to detect the boundary of a metric space, the potential use cases of magnitude remain under-explored. In this work, we investigate the properties of the magnitude on images, an important data modality in many machine learning applications. By endowing each individual images with its own metric space, we are able to define the concept of magnitude on images and analyse the individual contribution of each pixel with the magnitude vector. In particular, we theoretically show that the previously known properties of boundary detection translate to edge detection abilities in images. Furthermore, we demonstrate practical use cases of magnitude for machine learning applications and propose a novel magnitude model that consists of a computationally efficient magnitude computation and a learnable metric. By doing so, we address the computational hurdle that used to make magnitude impractical for many applications and open the way for the adoption of magnitude in machine learning research.