Compressibility Barriers to Neighborhood-Preserving Data Visualizations

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, $d = \Theta(\log n)$ is both necessary and sufficient for neighbor preservation. Even sparse regular graphs, which represent more restricted neighborhood connectivity structures, typically require $d= \Omega(\log n / \log\log n)$. The landscape changes dramatically when embedding into normed spaces: general graphs become exponentially harder to embed, requiring $d=\Omega(n)$, 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 $d=\Omega(\log n)$, even when the cluster structure is salient. These results challenge the aspiration that constant-dimensional visualizations can faithfully preserve neighborhood structure.