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 magnitude measure


The magnitude vector of images

arXiv.org Artificial Intelligence

The topology community has recently invested much effort in studying a newly introduced quantity called magnitude [1]. While it originates from category theory, where it can be seen as a generalisation of the Euler characteristic to metric spaces, the magnitude of a metric space is most intuitively understood as an attempt to measure the effective size of a metric space [2]. As a descriptive scalar, this quantity extends the set of other well known descriptors such as the rank, diameter or dimension. However, unlike those descriptors, the properties and potential use cases of magnitude are still under-explored. Because the metric space structure of datasets is a natural object of study when it comes to the understanding of fundamental machine learning concepts such as regularization, magnitude appears like a promising and powerful concept in machine learning: next to its abilities to describe the metric space of whole datasets, the magnitude can also be studied at the sample level, by considering each sample as its own metric space. Following this line of thought, magnitude vectors were introduced as a way to characterise the contribution of each data sample to the overall magnitude of the dataset, such that the sum of the elements of the magnitude vector amounts to the magnitude. This allowed to assess the individual contribution of each data point and their relative connectivity in the whole dataset. Indeed, magnitude vectors have been shown to detect boundaries of metric spaces, with boundary points exhibiting larger contributions to the magnitude [3].