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.

翻译：在多大程度上可以将高维数据可视化于二维或三维图中？我们将此问题重新表述为：将n顶点图（表示输入点的邻域结构）嵌入到低倍增维度d的度量空间中，使得邻近点保持接近而非邻近点保持远离。这种邻域保持概念可理解为比近似等距更弱的嵌入约束，但在所需最小维度随点数变化的尺度关系上具有相似要求。我们证明对于绝大多数图，d = Θ(log n) 是邻域保持既必要又充分的条件。即使是表示更受限邻域连通结构的稀疏正则图，通常也需要d = Ω(log n / log log n)。当嵌入赋范空间时，情况发生显著变化：一般图变得指数级难以嵌入，需要d = Ω(n)，而稀疏正则图仍允许d = O(log n)。最后，我们研究这些结果对具有内在聚类结构数据可视化的启示。我们证明基于n个点上k个簇的植入分区模型生成的图通常需要d = Ω(log n)，即使聚类结构非常显著。这些结果对恒定维度可视化能够忠实保持邻域结构的期望提出了挑战。