Multi-dimensional Scaling (MDS) is a family of methods for embedding pair-wise dissimilarities between $n$ objects into low-dimensional space. MDS is widely used as a data visualization tool in the social and biological sciences, statistics, and machine learning. We study the Kamada-Kawai formulation of MDS: given a set of non-negative dissimilarities $\{d_{i,j}\}_{i , j \in [n]}$ over $n$ points, the goal is to find an embedding $\{x_1,\dots,x_n\} \subset \mathbb{R}^k$ that minimizes \[ \text{OPT} = \min_{x} \mathbb{E}_{i,j \in [n]} \left[ \left(1-\frac{\|x_i - x_j\|}{d_{i,j}}\right)^2 \right] \] Despite its popularity, our theoretical understanding of MDS is extremely limited. Recently, Demaine, Hesterberg, Koehler, Lynch, and Urschel (arXiv:2109.11505) gave the first approximation algorithm with provable guarantees for Kamada-Kawai, which achieves an embedding with cost $\text{OPT} +\epsilon$ in $n^2 \cdot 2^{\tilde{\mathcal{O}}(k \Delta^4 / \epsilon^2)}$ time, where $\Delta$ is the aspect ratio of the input dissimilarities. In this work, we give the first approximation algorithm for MDS with quasi-polynomial dependency on $\Delta$: for target dimension $k$, we achieve a solution with cost $\mathcal{O}(\text{OPT}^{ \hspace{0.04in}1/k } \cdot \log(\Delta/\epsilon) )+ \epsilon$ in time $n^{ \mathcal{O}(1)} \cdot 2^{\tilde{\mathcal{O}}( k^2 (\log(\Delta)/\epsilon)^{k/2 + 1} ) }$. Our approach is based on a novel analysis of a conditioning-based rounding scheme for the Sherali-Adams LP Hierarchy. Crucially, our analysis exploits the geometry of low-dimensional Euclidean space, allowing us to avoid an exponential dependence on the aspect ratio $\Delta$. We believe our geometry-aware treatment of the Sherali-Adams Hierarchy is an important step towards developing general-purpose techniques for efficient metric optimization algorithms.

翻译：多维缩放（MDS）是一类将$n$个对象间的成对相异性嵌入低维空间的方法。MDS被广泛用于社会科学、生物学、统计学和机器学习中的可视化工具。我们研究Kamada-Kawai提出的MDS形式化：给定$n$个点上的非负相异性度量$\{d_{i,j}\}_{i , j \in [n]}$，目标是找到嵌入$\{x_1,\dots,x_n\} \subset \mathbb{R}^k$，使得\[ \text{OPT} = \min_{x} \mathbb{E}_{i,j \in [n]} \left[ \left(1-\frac{\|x_i - x_j\|}{d_{i,j}}\right)^2 \right] \]最小化。尽管该方法应用广泛，但我们对MDS的理论理解极为有限。最近，Demaine、Hesterberg、Koehler、Lynch和Urschel（arXiv:2109.11505）首次为Kamada-Kawai方法提供了具有可证明保证的近似算法，该算法在$n^2 \cdot 2^{\tilde{\mathcal{O}}(k \Delta^4 / \epsilon^2)}$时间内实现成本为$\text{OPT} +\epsilon$的嵌入，其中$\Delta$是输入相异性度量的长宽比。本文首次提出对$\Delta$具有拟多项式依赖的MDS近似算法：对于目标维度$k$，我们在$n^{ \mathcal{O}(1)} \cdot 2^{\tilde{\mathcal{O}}( k^2 (\log(\Delta)/\epsilon)^{k/2 + 1} ) }$时间内实现成本为$\mathcal{O}(\text{OPT}^{ \hspace{0.04in}1/k } \cdot \log(\Delta/\epsilon) )+ \epsilon$的解。该方法基于对Sherali-Adams线性规划层次结构中基于条件化舍入方案的新颖分析。关键突破在于，我们利用低维欧氏空间的几何特性，避免了算法对长宽比$\Delta$的指数依赖。我们相信这种对Sherali-Adams层次结构的几何感知处理方法，是开发高效度量优化算法通用技术的重要一步。