@article{Adamer24a,
abstract = {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 image 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 one computational
hurdle that used to make magnitude impractical for
many applications and open the way for the adoption
of magnitude in machine learning research.},
archiveprefix = {arXiv},
author = {Adamer, Michael F. and De Brouwer, Edward and O'Bray, Leslie and Rieck, Bastian},
author+an = {4=highlight},
doi = {10.1007/s41468-024-00182-9},
eprint = {2110.15188},
journal = {Journal of Applied and Computational Topology},
number = {3},
pages = {447--473},
primaryclass = {cs.LG},
title = {The magnitude vector of images},
volume = {8},
year = {2024},
}