Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Representer Theorems for Metric and Preference Learning: Geometric Insights and Algorithms

Peyman Morteza

TL;DR

The paper introduces a unified Hilbert-space framework for metric and preference learning by defining a space of generalized Mahalanobis inner products \\mathcal{F}_{\\mathcal{H}} and deriving a novel representer theorem for simultaneous metric and preference learning, plus a simple representer theorem for metric learning from triplet comparisons. By leveraging regularization with the induced inner-product norm, the authors reduce infinite-dimensional problems to finite-dimensional ones and show how RKHS kernelization yields finite kernel representations, enabling scalable nonlinear algorithms. They develop kernelized algorithms that express solutions in terms of kernel terms and Gram matrices, and validate them on synthetic and real rank-inference benchmarks where the approach competitively outperforms baselines including vanilla ideal-point methods. The work provides both theoretical advances (new representer theorems) and practical algorithms with strong empirical performance, and it supplies code for reproducibility. Overall, the framework broadens the toolkit for metric and preference learning in nonlinear, kernelized settings with broad applicability to ranking and recommendation tasks.

Abstract

We develop a mathematical framework to address a broad class of metric and preference learning problems within a Hilbert space. We obtain a novel representer theorem for the simultaneous task of metric and preference learning. Our key observation is that the representer theorem for this task can be derived by regularizing the problem with respect to the norm inherent in the task structure. For the general task of metric learning, our framework leads to a simple and self-contained representer theorem and offers new geometric insights into the derivation of representer theorems for this task. In the case of Reproducing Kernel Hilbert Spaces (RKHSs), we illustrate how our representer theorem can be used to express the solution of the learning problems in terms of finite kernel terms similar to classical representer theorems. Lastly, our representer theorem leads to a novel nonlinear algorithm for metric and preference learning. We compare our algorithm against challenging baseline methods on real-world rank inference benchmarks, where it achieves competitive performance. Notably, our approach significantly outperforms vanilla ideal point methods and surpasses strong baselines across multiple datasets. Code available at: https://github.com/PeymanMorteza/Metric-Preference-Learning-RKHS

Representer Theorems for Metric and Preference Learning: Geometric Insights and Algorithms

TL;DR

The paper introduces a unified Hilbert-space framework for metric and preference learning by defining a space of generalized Mahalanobis inner products \\mathcal{F}_{\\mathcal{H}} and deriving a novel representer theorem for simultaneous metric and preference learning, plus a simple representer theorem for metric learning from triplet comparisons. By leveraging regularization with the induced inner-product norm, the authors reduce infinite-dimensional problems to finite-dimensional ones and show how RKHS kernelization yields finite kernel representations, enabling scalable nonlinear algorithms. They develop kernelized algorithms that express solutions in terms of kernel terms and Gram matrices, and validate them on synthetic and real rank-inference benchmarks where the approach competitively outperforms baselines including vanilla ideal-point methods. The work provides both theoretical advances (new representer theorems) and practical algorithms with strong empirical performance, and it supplies code for reproducibility. Overall, the framework broadens the toolkit for metric and preference learning in nonlinear, kernelized settings with broad applicability to ranking and recommendation tasks.

Abstract

We develop a mathematical framework to address a broad class of metric and preference learning problems within a Hilbert space. We obtain a novel representer theorem for the simultaneous task of metric and preference learning. Our key observation is that the representer theorem for this task can be derived by regularizing the problem with respect to the norm inherent in the task structure. For the general task of metric learning, our framework leads to a simple and self-contained representer theorem and offers new geometric insights into the derivation of representer theorems for this task. In the case of Reproducing Kernel Hilbert Spaces (RKHSs), we illustrate how our representer theorem can be used to express the solution of the learning problems in terms of finite kernel terms similar to classical representer theorems. Lastly, our representer theorem leads to a novel nonlinear algorithm for metric and preference learning. We compare our algorithm against challenging baseline methods on real-world rank inference benchmarks, where it achieves competitive performance. Notably, our approach significantly outperforms vanilla ideal point methods and surpasses strong baselines across multiple datasets. Code available at: https://github.com/PeymanMorteza/Metric-Preference-Learning-RKHS
Paper Structure (34 sections, 26 theorems, 101 equations, 4 figures, 2 tables, 2 algorithms)

This paper contains 34 sections, 26 theorems, 101 equations, 4 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Metric and Preference Learning Problem in Euclidean Spaces
  3. Simultaneous Metric and Preference Learning from Pairwise Comparisons
  4. Learning Metrics from Triplet Comparisons
  5. Space of Generalized Mahalanobis Inner Products
  6. Representer Theorems for Metric and Preference Learning
  7. Metric and Preference Learning Problem in Hilbert Spaces
  8. A Representer Theorem for Simultaneous Metric and Preference Learning
  9. An Illustrative Example
  10. A Representer Theorem for Metric Learning from Triplet Comparison
  11. Kernelized Algorithms for Metric and Preference Learning
  12. Eliminating Basis Dependence via the Gram Matrix
  13. Kernelized Algorithms for Metric and Preference Learning
  14. Experiments
  15. Experiments of Synthetic Data
  16. ...and 19 more sections

Key Result

Proposition 1

Let $A\in \mathcal{F}_{\mathcal{H}}$, then, defines an inner product on $\mathcal{H}$ that is equivalent to $\langle \cdot,\cdot\rangle_{\mathcal{H}}$. Conversely, let $g(\cdot,\cdot)$ be an inner product on $\mathcal{H}$ equivalent to $\langle \cdot,\cdot\rangle_{\mathcal{H}}$ then there exist a unique $A\in \mathcal{F}_{\mathcal{H}}$ such t

Figures (4)

  • Figure 1: Illustration of the variation of preference loss with respect to the underlying geometry. When projecting along the $\langle \cdot,\cdot\rangle_{\mathcal{H}}$, the loss value changes along the projection line. However, when projecting along the line induced by $\langle \cdot,\cdot\rangle_{A}$, the loss remains constant.
  • Figure 2: The data distribution is supported along two concentric circles with Gaussian noise of variance $0.4$. In this setup, the left portion of the larger circle and the right portion of the smaller circle are labeled as $0$, while the remaining regions are labeled as $1$. This is illustrated in the figure above, with label $0$ depicted in red and label $1$ in blue. For our experiments, we assume that the user prefers all points labeled $0$ over those labeled $1$, and the objective is to search for the ideal point $u$ and the PSD matrix $A$ that align with these preferences.
  • Figure : Kernelized Ideal Point Algorithm: Training
  • Figure : Kernelized Ideal Point Algorithm: Testing

Theorems & Definitions (79)

  • Definition 1: mahalanobis1936generalized
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 2: Space of generalized Mahalanobis inner products
  • Remark 4
  • Remark 4
  • Remark 5
  • Proposition 1
  • proof
  • ...and 69 more