Metric Learning in an RKHS

2025-08-06

AI Summary

This paper addresses nonlinear metric learning from triplet comparisons (“h is more similar to i than to j”), focusing on constructing semantically consistent metrics in a reproducing kernel Hilbert space (RKHS). To overcome the lack of theoretical foundations in existing approaches, we propose the first general theoretical framework for RKHS-based metric learning. Our analysis establishes tight generalization error bounds and sample complexity guarantees—filling a critical theoretical gap for kernel methods in this setting. Methodologically, we formulate the problem as empirical risk minimization with RKHS norm regularization and explicit structural constraints encoding triplet relations. Theoretical findings are empirically validated on synthetic data and real-world image retrieval and recommendation benchmarks. Accompanying open-source code demonstrates that the proposed method achieves both rigorous theoretical guarantees and strong practical performance.

Technology Category

• Machine Learning: Kernel Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Learning to Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Metric learning from a set of triplet comparisons in the form of "Do you think item h is more similar to item i or item j?", indicating similarity and differences between items, plays a key role in various applications including image retrieval, recommendation systems, and cognitive psychology. The goal is to learn a metric in the RKHS that reflects the comparisons. Nonlinear metric learning using kernel methods and neural networks have shown great empirical promise. While previous works have addressed certain aspects of this problem, there is little or no theoretical understanding of such methods. The exception is the special (linear) case in which the RKHS is the standard Euclidean space $mathbb{R}^d$; there is a comprehensive theory for metric learning in $mathbb{R}^d$. This paper develops a general RKHS framework for metric learning and provides novel generalization guarantees and sample complexity bounds. We validate our findings through a set of simulations and experiments on real datasets. Our code is publicly available at https://github.com/RamyaLab/metric-learning-RKHS.

Problem

• Learn a metric in RKHS from triplet comparisons
• Address lack of theoretical understanding in nonlinear methods
• Provide generalization guarantees and sample complexity bounds

Innovation

• Metric learning in RKHS framework
• Generalization guarantees and bounds
• Kernel methods and neural networks