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Abstract

Low-dimensional embeddings (LDEs) of high-dimensional data are ubiquitous in
science and engineering. They allow us to quickly understand the main
properties of the data, identify outliers and processing errors, and inform the
next steps of data analysis. As such, LDEs have to be faithful to the original
high-dimensional data, i.e., they should represent the relationships that are
encoded in the data, both at a local as well as global scale. The current
generation of LDE approaches focus on reconstructing local distances between
any pair of samples correctly, often out-performing traditional approaches
aiming at all distances. For these approaches, global relationships are,
however, usually strongly distorted, often argued to be an inherent trade-off
between local and global structure learning for embeddings. We suggest a new
perspective on LDE learning, reconstructing angles between data points. We show
that this approach, Mercat, yields good reconstruction across a diverse set of
experiments and metrics, and preserve structures well across all scales.
Compared to existing work, our approach also has a simple formulation,
facilitating future theoretical analysis and algorithmic improvements.