A Survey on Metric Learning for Feature Vectors and Structured Data

TL;DR

This survey addresses the challenge of learning effective distance and similarity measures for both feature vectors and structured data. It surveys a broad spectrum of approaches, from classical Mahalanobis-distance methods (with convex, online, and multi-task variants) to nonlinear, local, histogram-based, and kernelized techniques, and extends into metric learning for edit distances on strings, trees, and graphs. A key contribution is the comprehensive taxonomy, highlighting trade-offs in learning paradigm, metric form, scalability, optimality, and dimensionality reduction, as well as generalization guarantees and semi-supervised/domain-adaptation extensions. The paper also identifies gaps—especially in theory beyond linear classification and in scalable methods for structured data—and outlines promising directions, including unsupervised metric learning, structure-aware approaches, and methods that adapt to changing data. Collectively, it provides a detailed roadmap for researchers and practitioners seeking to select and develop metric-learning methods across diverse data types and applications.

Abstract

The need for appropriate ways to measure the distance or similarity between data is ubiquitous in machine learning, pattern recognition and data mining, but handcrafting such good metrics for specific problems is generally difficult. This has led to the emergence of metric learning, which aims at automatically learning a metric from data and has attracted a lot of interest in machine learning and related fields for the past ten years. This survey paper proposes a systematic review of the metric learning literature, highlighting the pros and cons of each approach. We pay particular attention to Mahalanobis distance metric learning, a well-studied and successful framework, but additionally present a wide range of methods that have recently emerged as powerful alternatives, including nonlinear metric learning, similarity learning and local metric learning. Recent trends and extensions, such as semi-supervised metric learning, metric learning for histogram data and the derivation of generalization guarantees, are also covered. Finally, this survey addresses metric learning for structured data, in particular edit distance learning, and attempts to give an overview of the remaining challenges in metric learning for the years to come.

Figures (6)

• Figure 1: Illustration of metric learning applied to a face recognition task. For simplicity, images are represented as points in 2 dimensions. Pairwise constraints, shown in the left pane, are composed of images representing the same person (must-link, shown in green) or different persons (cannot-link, shown in red). We wish to adapt the metric so that there are fewer constraint violations (right pane). Images are taken from the Caltech Faces dataset.
• Figure 2: The common process in metric learning. A metric is learned from training data and plugged into an algorithm that outputs a predictor (e.g., a classifier, a regressor, a recommender system...) which hopefully performs better than a predictor induced by a standard (non-learned) metric.
• Figure 3: Five key properties of metric learning algorithms.
• Figure 4: The cone $\mathbb{S}^{2}_+$ of positive semi-definite 2x2 matrices of the form $\alpha\beta\beta\gamma$.
• Figure 5: The two-fold problem of generalization in metric learning. We may be interested in the generalization ability of the learned metric itself: can we say anything about its consistency on unseen data drawn from the same distribution? Furthermore, we may also be interested in the generalization ability of the predictor using that metric: can we relate its performance on unseen data to the quality of the learned metric?

Theorems & Definitions (2)

• Definition 1: Alphabet and string
• Definition 2: String edit distance