The magnitude of a metric space was recently established as a novel
invariant, providing a measure of the `effective size' of a space across
multiple scales. By capturing both geometrical and topological properties of
data, magnitude is poised to address challenges in unsupervised representation
learning tasks. We formalise a novel notion of dissimilarity between magnitude
functions of finite metric spaces and use them to derive a quality measure for
dimensionality reduction tasks. Our measure is provably stable under
perturbations of the data, can be efficiently calculated, and enables a
rigorous multi-scale comparison of embeddings. We show the utility of our
measure in an experimental suite that comprises different domains and tasks,
including the comparison of data visualisations.

度量空间的规模最近被确定为一种新型不变量，它可以在多个尺度上衡量空间的 ' 有效大小 '。通过捕捉数据的几何和拓扑特性，度量空间可以解决无监督表示学习任务中的挑战。我们形式化了有限度量空间的度量函数之间的新概念差异度，并用它们导出了一种用于降维任务的质量度量。我们的度量可以在数据扰动下保持稳定，计算效率高，并且可以对嵌入进行严格的多尺度比较。我们通过一个实验套件展示了我们度量的实用性，其中包括数据可视化的比较。