Gromov-Wasserstein Methods for Multi-View Relational Embedding and Clustering

Multi-View Clustering (MVC) has gained significant attention for its ability to leverage complementary information across diverse views. However, existing deep MVC methods often struggle with view-distribution entanglement during cross-view fusion, which hampers the quality of the shared latent space and leads to suboptimal Figures. To address this issue, we propose the Generalized Multi-view Auto-Encoder (GMAE), a framework designed to preserve cross-view complementarity through disentangled representation learning. Specifically, GMAE employs dual-path autoencoders to decouple source features into view-specific and view-common embeddings, facilitating the discovery of clearer clustering structures. We further construct cross-view adversarial discriminators to guide view-specific encoders in capturing more discriminative features. By strategically modulating mutual information, GMAE effectively aligns distributions and prevents representation collapse, ensuring the generation of robust, non-trivial embeddings. Comprehensive experiments on 13 benchmark datasets demonstrate that GMAE consistently outperforms state-of-the-art methods in both complete and incomplete MVC tasks. Our code implementation is available at the repository: https://github.com/obananas/GMAE.

Incomplete Multi-View Clustering (IMVC) faces the challenge of learning discriminative representations from fragmentary observations while maintaining robustness against missing views. However, prevalent Euclidean-based methods suffer from a geometric mismatch when modeling real-world data with intrinsic hierarchies, leading to semantic blurring where representations drift towards spatially proximal but semantically distinct neighbors. To bridge this gap, we propose HERL, a Hyperbolic Enhanced Representation Learning framework for IMVC. Operating within the Poincaré ball, HERL constructs a structure-aware latent space to enhance representation learning. Specifically, we design a dual-constraint hyperbolic contrastive mechanism optimizing: an angular-based loss to preserve semantic identity via directional alignment, and a distance-based loss to enforce hierarchical compactness. Furthermore, a hyperbolic prototype head is introduced to rectify global structural drift by aligning cross-view hierarchy-aware prototype distributions. Consequently, HERL disentangles fine-grained semantic correlations to sharpen cluster boundaries and imposes geometric constraints to rectify the data recovery process. Extensive experimental results demonstrate that HERL consistently outperforms state-of-the-art approaches.

Gromov--Wasserstein (GW) distances compare graphs, shapes, and point clouds through internal distances, without requiring a common coordinate system. This invariance is powerful, but discrete GW is a nonconvex quadratic optimal transport problem and is difficult to estimate at scale. We propose \emph{Distance-Matrix Wasserstein} (DMW), a hierarchy of Wasserstein statistics comparing laws of random finite distance matrices. Rather than optimizing a global point-level alignment, DMW samples $n$ points from each space, records their pairwise distances, and transports the resulting matrix laws. We prove that DMW is a relaxation and lower bound of GW, and establish a reverse approximation inequality: the GW--DMW gap is controlled by the Wasserstein error of approximating each original measure with $n$ samples. Hence population DMW converges to GW as sampled subspaces become dense. We further give finite-sample bounds, including intrinsic-dimensional rates that depend on the data manifold rather than the ambient matrix dimension $\binom n2$. For scalable computation, we introduce sliced and multi-scale DMW; for $p=1$, the sliced multi-scale dissimilarity yields positive-definite exponential kernels. Experiments on synthetic metric spaces, scalability benchmarks, graph classification, and two-sample testing validate the theory and demonstrate an interpretable GW-style proxy for structural comparison.