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

Learning low-dimensional representations from multi-view relational data is challenging when underlying geometries differ across views. We propose Bary-GWMDS, a Gromov-Wasserstein-based method that operates directly on distance matrices to learn a consensus embedding preserving shared relational structure. By leveraging intrinsic distances, the approach naturally handles nonlinear distortions across views. We also introduce Mean-GWMDS-C, a clustering-oriented formulation that averages distance matrices and learns reduced-support representations via a consensus Gromov-Wasserstein transport. Experiments on synthetic and real-world datasets show that the proposed framework yields stable and geometrically meaningful embeddings.

Gromov-Wasserstein multidimensional scaling (GW-MDS) learns low-dimensional representations from relational data but remains transductive, providing no explicit mapping for unseen samples. We introduce an inductive framework based on barycentric distillation. A GW-MDS teacher learns a latent support and an optimal transport plan from the training data, and barycentric projection converts the resulting coupling into sample-aligned targets. A neural student then learns an explicit out-of-sample mapping, avoiding additional relational-matrix construction and GW optimization at inference. We formulate the approach for single-view data and extend it to Mean-GWMDS and Multi-GWMDS teachers through consensus and selected-projection targets learned by a multi-view student with view-specific encoders. We also investigate a direct neural baseline trained solely with a GW objective. Experiments on synthetic and real-world data using Euclidean, geodesic, and cosine relations show that the distilled models preserve the teacher geometry on unseen samples and consistently outperform direct neural GW training in sample-indexed relational preservation. These results establish barycentric projection as an effective bridge between transductive GW embeddings and inductive neural mappings.

Multi-view learning jointly exploits multiple complementary representations of the same data and has become increasingly important in machine learning. However, the Python ecosystem lacks actively maintained, unified tooling for end-to-end multi-view workflows. In this paper, we present polyview, a Python package that provides tools for multi-view embedding, clustering, fusion, and view augmentation, as well as for handling incomplete views, all compatible with scikit-learn. The library offers a unified interface for composing heterogeneous multi-view workflows, including seamless transitions between multi-view and single-view stages. It is built around a core set of classes and utilities that enable composition of different methods and straightforward implementation of new ones. We illustrate the package on five real multi-view datasets and compare its components based on canonical correlation analysis with those of two established libraries. polyview aims to be both a practical toolkit for benchmarking and prototyping multi-view methods and a foundation for future research and development in this area.