@online{Fischer_2406.09876,
TITLE = {Sailing in High-dimensional Spaces: {L}ow-dimensional Embeddings through Angle Preservation},
AUTHOR = {Fischer, Jonas and Ma, Rong},
LANGUAGE = {eng},
URL = {https://arxiv.org/abs/2406.09876},
EPRINT = {2406.09876},
EPRINTTYPE = {arXiv},
YEAR = {2024},
MARGINALMARK = {$\bullet$},
ABSTRACT = {Low-dimensional embeddings (LDEs) of high-dimensional data are ubiquitous in<br>science and engineering. They allow us to quickly understand the main<br>properties of the data, identify outliers and processing errors, and inform the<br>next steps of data analysis. As such, LDEs have to be faithful to the original<br>high-dimensional data, i.e., they should represent the relationships that are<br>encoded in the data, both at a local as well as global scale. The current<br>generation of LDE approaches focus on reconstructing local distances between<br>any pair of samples correctly, often out-performing traditional approaches<br>aiming at all distances. For these approaches, global relationships are,<br>however, usually strongly distorted, often argued to be an inherent trade-off<br>between local and global structure learning for embeddings. We suggest a new<br>perspective on LDE learning, reconstructing angles between data points. We show<br>that this approach, Mercat, yields good reconstruction across a diverse set of<br>experiments and metrics, and preserve structures well across all scales.<br>Compared to existing work, our approach also has a simple formulation,<br>facilitating future theoretical analysis and algorithmic improvements.<br>},
}