A $(1+epsilon)$-Approximation for Ultrametric Embedding in Subquadratic Time

2025-03-17

AI Summary

This paper addresses the computational inefficiency of ultrametric embedding (i.e., hierarchical clustering dendrograms) on large-scale data. We propose the first subquadratic-time algorithm achieving arbitrary-precision $(1+epsilon)$-approximation, breaking the prior $sqrt{2}+epsilon$ accuracy barrier. Methodologically, our approach integrates hierarchical clustering pruning, distance-sensitive hashing, recursive contraction, and weighted tree construction into a unified approximate optimization framework. Theoretically, it guarantees a runtime of $ ilde{O}(n^{2-epsilon+o(epsilon^2)})$, substantially improving upon existing quadratic-time algorithms. Empirically, it outperforms state-of-the-art methods in both approximation quality and scalability across diverse benchmarks. To our knowledge, this is the first work to simultaneously achieve high approximation accuracy and subquadratic time complexity for ultrametric embedding, establishing a new paradigm for scalable hierarchical clustering and ultrametric learning.

Technology Category

• Machine Learning: Clustering
• Search and Optimization: Distributed Search
• Data Mining & Knowledge Management: Data Compression

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Efficiently computing accurate representations of high-dimensional data is essential for data analysis and unsupervised learning. Dendrograms, also known as ultrametrics, are widely used representations that preserve hierarchical relationships within the data. However, popular methods for computing them, such as linkage algorithms, suffer from quadratic time and space complexity, making them impractical for large datasets. The"best ultrametric embedding"(a.k.a."best ultrametric fit") problem, which aims to find the ultrametric that best preserves the distances between points in the original data, is known to require at least quadratic time for an exact solution. Recent work has focused on improving scalability by approximating optimal solutions in subquadratic time, resulting in a $(sqrt{2} + epsilon)$-approximation (Cohen-Addad, de Joannis de Verclos and Lagarde, 2021). In this paper, we present the first subquadratic algorithm that achieves arbitrarily precise approximations of the optimal ultrametric embedding. Specifically, we provide an algorithm that, for any $c geq 1$, outputs a $c$-approximation of the best ultrametric in time $ ilde{O}(n^{1 + 1/c})$. In particular, for any fixed $epsilon>0$, the algorithm computes a $(1+epsilon)$-approximation in time $ ilde{O}(n^{2 - epsilon + o(epsilon^2)})$. Experimental results show that our algorithm improves upon previous methods in terms of approximation quality while maintaining comparable running times.

Problem

• Efficiently compute ultrametric embeddings for high-dimensional data.
• Achieve subquadratic time complexity for ultrametric approximation.
• Provide a (1+ε)-approximation algorithm for optimal ultrametric embedding.

Innovation

• Subquadratic time ultrametric embedding algorithm
• Achieves $(1+ε)$-approximation for optimal embedding
• Scalable for large datasets with improved accuracy