Metric Embeddings Beyond Bi-Lipschitz Distortion via Sherali-Adams

TL;DR

The paper develops a polynomial-time framework for approximating the Kamada-Kawai metric embedding objective (MDS) with nontrivial dependence on the input aspect ratio $\Delta$. It discretizes the target space, then solves a high-level Sherali-Adams LP and applies a conditioning-based rounding to produce an embedding in $\mathbb{R}^k$ whose KK cost is bounded by $\tilde{O}(\log \Delta)\cdot OPT^{Ω(1)} + \varepsilon$, avoiding exponential dependence on $\Delta$. The key technical advance is a geometry-aware analysis of conditional rounding that controls variances and distances via quantiles relative to an optimal embedding, plus rigorous discretization and dimension-reduction arguments. The results establish the first nontrivial approximation for MDS with super-logarithmic aspect ratio in polynomial time and open avenues for data-dependent embedding methods beyond bi-Lipschitz guarantees. Practical impact includes improved, theoretically-grounded MDS approaches for dimension reduction and visualization that handle large aspect ratios more efficiently than prior methods.

Abstract

Metric embeddings are a widely used method in algorithm design, where generally a ``complex'' metric is embedded into a simpler, lower-dimensional one. Historically, the theoretical computer science community has focused on bi-Lipschitz embeddings, which guarantee that every pairwise distance is approximately preserved. In contrast, alternative embedding objectives that are commonly used in practice avoid bi-Lipschitz distortion; yet these approaches have received comparatively less study in theory. In this paper, we focus on Multi-dimensional Scaling (MDS), where we are given a set of non-negative dissimilarities $\{d_{i,j}\}_{i,j\in [n]}$ over $n$ points, and the goal is to find an embedding $\{x_1,\dots,x_n\} \subset R^k$ that minimizes $$\textrm{OPT}=\min_{x}\mathbb{E}_{i,j\in [n]}\left(1-\frac{\|x_i - x_j\|}{d_{i,j}}\right)^2.$$ Despite its popularity, our theoretical understanding of MDS is extremely limited. Recently, Demaine et. al. (arXiv:2109.11505) gave the first approximation algorithm with provable guarantees for this objective, which achieves an embedding in constant dimensional Euclidean space with cost $\textrm{OPT} +ε$ in $n^2\cdot 2^{\textrm{poly}(Δ/ε)}$ time, where $Δ$ is the aspect ratio of the input dissimilarities. For metrics that admit low-cost embeddings, $Δ$ scales polynomially in $n$. In this work, we give the first approximation algorithm for MDS with quasi-polynomial dependency on $Δ$: for constant dimensional Euclidean space, we achieve a solution with cost $O(\log Δ)\cdot \textrm{OPT}^{Ω(1)}+ε$ in time $n^{O(1)} \cdot 2^{\text{poly}((\log(Δ)/ε))}$. Our algorithms are based on a novel geometry-aware analysis of a conditional rounding of the Sherali-Adams LP Hierarchy, allowing us to avoid exponential dependency on the aspect ratio, which would typically result from this rounding.

Figures (4)

• Figure 1: Variance of $x_i$ after conditioning on $x_j$ is at most $\mathop{\mathrm{\tilde{\mathbb{E}}}}\nolimits \|x_i - x_j\|^2$.
• Figure 2: Example -- uniformly-spaced metric $x_1*,\ldots,x_n^*$
• Figure 3: Illustration of an input instance where MDS preserves meaningful cluster structure but PCA does not.
• Figure 4: Illustration of the optimal MDS mapping for this instance. The red points are the points previously at $(i, \pm \varepsilon)$ that which were mapped onto the line by MDS, whereas PCA would collapse each pair of red points onto their central black point.

Theorems & Definitions (34)

• Theorem 1.1: Approximation schemes scaling exponentially in aspect ratio demaine2021multidimensional
• Theorem 1.2: Main theorem, qualitative version
• Remark 1.3: Comparison with demaine2021multidimensional
• Lemma 2.2: Key lemma on $k$-dimensional discrete metrics
• Theorem 5.1: Main Theorem
• Remark 5.2
• Lemma 5.4: Aspect ratio of the target space
• Definition 5.5: Pseudo-Deviation of an Embedding
• Lemma 5.7: Pseudodeviation Reduction via Quantiles
• Lemma 5.8: Rounded cost on a small set of pairs