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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2508.07119 |
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| _version_ | 1866911378318557184 |
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| author | Snoeck, Szymon Bergam, Noah Verma, Nakul |
| author_facet | Snoeck, Szymon Bergam, Noah Verma, Nakul |
| contents | To what extent is it possible to visualize high-dimensional data in two- or three-dimensional plots? We reframe this question in terms of embedding $n$-vertex graphs (representing the neighborhood structure of the input points) into metric spaces of low doubling dimension $d$ in such a way that keeps neighbors close and non-neighbors far. This notion of neighbor preservation can be understood as a considerably weaker embedding constraint than near-isometry, yet it is similarly as demanding in terms of how the minimum required dimension scales with the number of points. We show that for an overwhelming fraction of graphs, $d = Θ(\log n)$ is both necessary and sufficient for neighbor preservation. Even sparse regular graphs, which represent more restricted neighborhood connectivity structures, typically require $d= Ω(\log n / \log\log n)$. The landscape changes dramatically when embedding into normed spaces: general graphs become exponentially harder to embed, requiring $d=Ω(n)$, while sparse regular graphs continue to admit $d = O(\log n)$. Finally, we study the implications of these results for visualizing data with intrinsic cluster structure. We show that graphs produced from a planted partition model with $k$ clusters on $n$ points typically require $d=Ω(\log n)$, even when the cluster structure is salient. These results challenge the aspiration that constant-dimensional visualizations can faithfully preserve neighborhood structure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07119 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Compressibility Barriers to Neighborhood-Preserving Data Visualizations Snoeck, Szymon Bergam, Noah Verma, Nakul Computational Geometry Metric Geometry To what extent is it possible to visualize high-dimensional data in two- or three-dimensional plots? We reframe this question in terms of embedding $n$-vertex graphs (representing the neighborhood structure of the input points) into metric spaces of low doubling dimension $d$ in such a way that keeps neighbors close and non-neighbors far. This notion of neighbor preservation can be understood as a considerably weaker embedding constraint than near-isometry, yet it is similarly as demanding in terms of how the minimum required dimension scales with the number of points. We show that for an overwhelming fraction of graphs, $d = Θ(\log n)$ is both necessary and sufficient for neighbor preservation. Even sparse regular graphs, which represent more restricted neighborhood connectivity structures, typically require $d= Ω(\log n / \log\log n)$. The landscape changes dramatically when embedding into normed spaces: general graphs become exponentially harder to embed, requiring $d=Ω(n)$, while sparse regular graphs continue to admit $d = O(\log n)$. Finally, we study the implications of these results for visualizing data with intrinsic cluster structure. We show that graphs produced from a planted partition model with $k$ clusters on $n$ points typically require $d=Ω(\log n)$, even when the cluster structure is salient. These results challenge the aspiration that constant-dimensional visualizations can faithfully preserve neighborhood structure. |
| title | Compressibility Barriers to Neighborhood-Preserving Data Visualizations |
| topic | Computational Geometry Metric Geometry |
| url | https://arxiv.org/abs/2508.07119 |