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Structure-Preserving Multi-View Embedding Using Gromov-Wasserstein Optimal Transport

Rafael Pereira Eufrazio, Eduardo Fernandes Montesuma, Charles Casimiro Cavalcante

Abstract

Multi-view data analysis seeks to integrate multiple representations of the same samples in order to recover a coherent low-dimensional structure. Classical approaches often rely on feature concatenation or explicit alignment assumptions, which become restrictive under heterogeneous geometries or nonlinear distortions. In this work, we propose two geometry-aware multi-view embedding strategies grounded in Gromov-Wasserstein (GW) optimal transport. The first, termed Mean-GWMDS, aggregates view-specific relational information by averaging distance matrices and applying GW-based multidimensional scaling to obtain a representative embedding. The second strategy, referred to as Multi-GWMDS, adopts a selection-based paradigm in which multiple geometry-consistent candidate embeddings are generated via GW-based alignment and a representative embedding is selected. Experiments on synthetic manifolds and real-world datasets show that the proposed methods effectively preserve intrinsic relational structure across views. These results highlight GW-based approaches as a flexible and principled framework for multi-view representation learning.

Structure-Preserving Multi-View Embedding Using Gromov-Wasserstein Optimal Transport

Abstract

Multi-view data analysis seeks to integrate multiple representations of the same samples in order to recover a coherent low-dimensional structure. Classical approaches often rely on feature concatenation or explicit alignment assumptions, which become restrictive under heterogeneous geometries or nonlinear distortions. In this work, we propose two geometry-aware multi-view embedding strategies grounded in Gromov-Wasserstein (GW) optimal transport. The first, termed Mean-GWMDS, aggregates view-specific relational information by averaging distance matrices and applying GW-based multidimensional scaling to obtain a representative embedding. The second strategy, referred to as Multi-GWMDS, adopts a selection-based paradigm in which multiple geometry-consistent candidate embeddings are generated via GW-based alignment and a representative embedding is selected. Experiments on synthetic manifolds and real-world datasets show that the proposed methods effectively preserve intrinsic relational structure across views. These results highlight GW-based approaches as a flexible and principled framework for multi-view representation learning.

Paper Structure

This paper contains 13 sections, 13 equations, 5 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Optimal Transport
  4. Gromov–Wasserstein Distance
  5. Gromov–Wasserstein Multidimensional Scaling
  6. Multi-View Learning
  7. Proposed Methods
  8. Mean-GWMDS: Distance Averaging Strategy
  9. Multi-GWMDS: Selection-Based Strategy
  10. Experiments and Discussion
  11. Manifold experiments
  12. Real-World Experiments: Electricity Load Diagrams
  13. Conclusions and future directions

Figures (5)

  • Figure 1: Embeddings obtained on the S-curve dataset using different multi-view methods.
  • Figure 2: Embeddings obtained on the S-curve dataset using different multi-view strategies with geodesic distance.
  • Figure 3: Low-dimensional embeddings of the Electricity Load Diagrams (ELD) dataset obtained using different multi-view methods under geodesic distances.
  • Figure 4: Low-dimensional embeddings of the Electricity Load Diagrams (ELD) dataset obtained with Multi-GWMDS using geodesic distances. Each panel corresponds to an embedding induced by a different view.
  • Figure 5: Embeddings on the Electricity Load Diagrams dataset obtained with different multi-view methods. Multi-GWMDS and Mean-GWMDS use Euclidean distances and are contrasted with Euclidean and correlation-based baselines.

Theorems & Definitions (1)

  • Definition 1