The magnitude vector of images

Michael F Adamer, Leslie O'Bray, Edward De Brouwer, Bastian Rieck, Karsten Borgwardt

Keywords: magnitude, magnitude vector, edge detection, adversarial robustness, metric space, algebraic topology

Abstract: The magnitude of a finite metric space is a recently-introduced invariant quantity. Despite beneficial theoretical and practical properties, such as a general utility for outlier detection, and a close connection to Laplace radial basis kernels, magnitude has received little attention by the machine learning community so far. In this work, we investigate the properties of magnitude on individual images, with each image forming its own metric space. We show that the known properties of outlier detection translate to edge detection in images and we give supporting theoretical justifications. In addition, we provide a proof of concept of its utility by using a novel magnitude layer to defend against adversarial attacks. Since naive magnitude calculations may be computationally prohibitive, we introduce an algorithm that leverages the regular structure of images to dramatically reduce the computational cost.

One-sentence Summary: We study the magnitude vector of images and highlight potential applications

Community Implementations: [ 2 code implementations](https://www.catalyzex.com/paper/arxiv:2110.15188/code)