A quasi-polynomial time algorithm for Multi-Dimensional Scaling via LP hierarchies

Multi-dimensional Scaling (MDS) is a family of methods for embedding an n-point metric into low-dimensional Euclidean space. We study the Kamada-Kawai formulation of MDS: given a set of non-negative dissimilarities {d_i,j}_i , j ∈ [n] over n points, the goal is to find an embedding {x_1,…,x_n}∈ℝ^k that minimizes OPT = min_x𝔼_i,j ∈ [n][ (1-x_i - x_j/d_i,j)^2 ] Kamada-Kawai provides a more relaxed measure of the quality of a low-dimensional metric embedding than the traditional bi-Lipschitz-ness measure studied in theoretical computer science; this is advantageous because strong hardness-of-approximation results are known for the latter, Kamada-Kawai admits nontrivial approximation algorithms. Despite its popularity, our theoretical understanding of MDS is limited. Recently, Demaine, Hesterberg, Koehler, Lynch, and Urschel (arXiv:2109.11505) gave the first approximation algorithm with provable guarantees for Kamada-Kawai in the constant-k regime, with cost OPT +ϵ in n^2 2^poly(Δ/ϵ) time, where Δ is the aspect ratio of the input. In this work, we give the first approximation algorithm for MDS with quasi-polynomial dependency on Δ: we achieve a solution with cost Õ(logΔ)OPT^Ω(1)+ϵ in time n^O(1)2^poly(log(Δ)/ϵ). 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. 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.