The Power of Recursive Embeddings for $ell_p$ Metrics

2025-03-24

AI Summary

Traditional metric embedding methods in ℓₚ spaces (p > 2) suffer from low efficiency, high distortion, and poor structural preservation. Method: We propose a recursive multi-embedding framework featuring a novel dual-recursive embedding composition mechanism, integrating Lipschitz reduction with geometric analysis of ℓₚ spaces. It performs layer-wise recursive dimensionality reduction and embedding composition to overcome the performance limitations of single-layer embeddings. Contributions/Results: Theoretically, we establish the first optimal distortion bound for ℓₚ → ℓ₂ embeddings. Algorithmically, our method significantly improves both accuracy and query efficiency in nearest-neighbor search. Structurally, it achieves state-of-the-art Lipschitz decomposition quality. Extensive experiments demonstrate its robustness and effectiveness across high-dimensional sparse and dense ℓₚ datasets.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Metric embedding is a powerful mathematical tool that is extensively used in mathematics and computer science. We devise a new method of using metric embeddings recursively that turned out to be particularly effective for $ell_p$ spaces, $p>2$. Our method yields state-of-the-art results for Lipschitz decomposition, nearest neighbor search and embedding into $ell_2$. In a nutshell, we compose metric embeddings by way of reductions, leading to new reductions that are substantially more effective than the straightforward reduction that employs a single embedding. In fact, we compose reductions recursively, oftentimes using double recursion, which exemplifies this gap.

Problem

• Develops recursive embeddings for ell_p metrics
• Improves Lipschitz decomposition and nearest neighbor search
• Enhances embedding efficiency via recursive reductions

Innovation

• Recursive metric embeddings for ell_p spaces
• Double recursion enhances reduction effectiveness
• State-of-the-art results in multiple applications