Metric $k$-clustering using only Weak Comparison Oracles

TL;DR

This paper introduces randomized algorithms for $k$-clustering that operate solely on weak comparison oracles, replacing exact distance queries with noisy relative comparisons, and achieves near-optimal clustering with low query complexity.

Contribution

It develops the first scalable clustering algorithms that use only noisy quadruplet comparison oracles, extending applicability to scenarios with limited or noisy distance information.

Findings

Achieves constant-factor approximation with $O(nk ext{polylog}(n))$ queries.

Improves to $(1+ ext{small }\varepsilon)$-approximation for bounded doubling dimension metrics.

Demonstrates integration of noisy, low-cost oracles like language models into clustering algorithms.

Abstract

Clustering is a fundamental primitive in unsupervised learning. However, classical algorithms for $k$-clustering (such as $k$-median and $k$-means) assume access to exact pairwise distances -- an unrealistic requirement in many modern applications. We study clustering in the \emph{Rank-model (R-model)}, where access to distances is entirely replaced by a \emph{quadruplet oracle} that provides only relative distance comparisons. In practice, such an oracle can represent learned models or human feedback, and is expected to be noisy and entail an access cost. Given a metric space with $n$ input items, we design randomized algorithms that, using only a noisy quadruplet oracle, compute a set of $O(k \cdot \mathsf{polylog}(n))$ centers along with a mapping from the input items to the centers such that the clustering cost of the mapping is at most constant times the optimum $k$-clustering cost. Our method achieves a query complexity of $O(n\cdot k \cdot \mathsf{polylog}(n))$ for arbitrary metric spaces and improves to $O((n+k^2) \cdot \mathsf{polylog}(n))$ when the underlying metric has bounded doubling dimension. When the metric has bounded doubling dimension we can further improve the approximation from constant to $1+\varepsilon$, for any arbitrarily small constant $\varepsilon\in(0,1)$, while preserving the same asymptotic query complexity. Our framework demonstrates how noisy, low-cost oracles, such as those derived from large language models, can be systematically integrated into scalable clustering algorithms.