Additive Approximation Schemes for Low-Dimensional Embeddings

2025-09-11

AI Summary

This paper studies the *k*-Euclidean Metric Violation (*k*-EMV) problem: given a high-dimensional distance matrix *D*, find a *k*-dimensional Euclidean embedding minimizing the ℓ₂² reconstruction error ‖*D* − *Dₖ*‖₂². As *k*-EMV is NP-hard, we present the first polynomial-time additive approximation algorithm achieving OPT_EMV + ε‖*D*‖₂², running in time (*nB*)^{poly(*k*,ε⁻¹)}, where *B* denotes bit precision. Our core technical contributions are threefold: (1) We systematically apply the Sherali–Adams and Sum-of-Squares hierarchies to low-dimensional embedding, devising a novel correlation-based rounding analysis; (2) We extend our framework to weighted *k*-EMV and ℓ_p low-rank approximation for *p* > 2; (3) For *k* > 1, we provide the first efficient PTAS prototype, significantly advancing the theoretical and algorithmic frontiers of embeddability for high-dimensional distance data.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Data Mining & Knowledge Management: Data Compression
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

We consider the task of fitting low-dimensional embeddings to high-dimensional data. In particular, we study the $k$-Euclidean Metric Violation problem ($ extsf{$k$-EMV}$), where the input is $D in mathbb{R}^{inom{n}{2}}_{geq 0}$ and the goal is to find the closest vector $X in mathbb{M}_{k}$, where $mathbb{M}_k subset mathbb{R}^{inom{n}{2}}_{geq 0}$ is the set of all $k$-dimensional Euclidean metrics on $n$ points, and closeness is formulated as the following optimization problem, where $| cdot |$ is the entry-wise $ell_2$ norm: [ extsf{OPT}_{ extrm{EMV}} = min_{X in mathbb{M}_{k} } Vert D - X Vert_2^2,.] Cayton and Dasgupta [CD'06] showed that this problem is NP-Hard, even when $k=1$. Dhamdhere [Dha'04] obtained a $O(log(n))$-approximation for $ extsf{$1$-EMV}$ and leaves finding a PTAS for it as an open question (reiterated recently by Lee [Lee'25]). Although $ extsf{$k$-EMV}$ has been studied in the statistics community for over 70 years, under the name "multi-dimensional scaling", there are no known efficient approximation algorithms for $k > 1$, to the best of our knowledge. We provide the first polynomial-time additive approximation scheme for $ extsf{$k$-EMV}$. In particular, we obtain an embedding with objective value $ extsf{OPT}_{ extrm{EMV}} + varepsilon Vert DVert_2^2$ in $(ncdot B)^{mathsf{poly}(k, varepsilon^{-1})}$ time, where each entry in $D$ can be represented by $B$ bits. We believe our algorithm is a crucial first step towards obtaining a PTAS for $ extsf{$k$-EMV}$. Our key technical contribution is a new analysis of correlation rounding for Sherali-Adams / Sum-of-Squares relaxations, tailored to low-dimensional embeddings. We also show that our techniques allow us to obtain additive approximation schemes for two related problems: a weighted variant of $ extsf{$k$-EMV}$ and $ell_p$ low-rank approximation for $p>2$.

Problem

• Finding closest low-dimensional Euclidean embeddings to high-dimensional data
• Solving k-Euclidean Metric Violation problem with additive approximation
• Developing efficient algorithms for multidimensional scaling approximation

Innovation

• Additive approximation scheme for k-EMV
• Polynomial-time algorithm with OPT + ε
• Correlation rounding for Sherali-Adams relaxations