Learning representations that capture both intrinsic data geometry and target-relevant structure remains a fundamental challenge, particularly in settings where data reduction must balance compression with predictive fidelity. While distributional reduction-encompassing joint clustering and dimensionality reduction-offers a principled way to summarize data, its supervised variants remain relatively under-explored, despite the importance of retaining task-relevant signal for downstream prediction and decision-making. We propose Supervised Distributional Reduction (SDR), an algorithm for learning target-aware representations by combining optimal transport with explicit dependence maximization. SDR builds on the Fused Gromov-Wasserstein (FGW) objective to align the relational structure of the input distribution with a set of representative points, while augmenting it with a direct dependence term that encourages the learned embeddings to capture predictive signal more explicitly. This results in compact representations that reflect both geometric structure and supervision. Beyond representation learning, SDR naturally induces a data-dependent, non-stationary geometry that can be leveraged for settings such as Gaussian Process (GP) modelling. By redefining distances through target-aware distributional alignment, SDR enables the construction of adaptive kernels that respond to local variations in both data geometry and supervision, offering an optimal transport-based perspective on non-stationary kernel design.
Prediction models trained under the source distribution do not generalize well to a different target distribution. A valid inference about an unseen data distribution must be anchored by the invariance of certain causal mechanisms that generate the source and target data, however, these structural invariances are non-identifiable from the source data alone. Under mild causal assumptions about the data, we show that the optimal prediction in the target is in fact partially identifiable by the source distribution. The result rests on a simple observation: In any domain, the optimal prediction can be factorized into what we call a pair of abduction and deduction maps, where the abduction map makes inference about some unobserved variables (possibly confounders) from the observed variables and the deduction map predicts the label using both the observed and inferred quantities. Access to large source data pins down the optimal prediction, thus constrains the valid abduction-deduction ensembles that produce it -- a non-identifiability that we call the abduction-deduction entanglement. To leverage this, we parameterize the constrained family using what we call a representation transplant, that is a specific linear transformation in the representation space that manipulates the abduction content of the representation while retaining the deduction component. Invariance of the causal mechanism generating the label implies existence of an invariant deduction map between source and target. Thus, we can search the space of plausible target distributions via a parametric transplant. We use this scheme in a learner-adversary game that, under an idealistic optimization, provably terminates with the learner having the minimax-optimal target prediction. Evaluations verify the theory, showing that the method is competitive in DG benchmarks.
A fundamental open question in self-supervised learning (SSL) is the explicit characterization of the optimal geometry of the learned representations. Recently, LeJEPA identified isotropic Gaussian embeddings as optimal for minimizing downstream prediction risk in Euclidean spaces. However, the corresponding problem for distributions supported on lower-dimensional manifolds, such as the hypersphere, remains unexplored. In this work, we demonstrate that extending this minimax analysis to smooth distributions on Riemannian manifolds fundamentally changes the optimal solution. We show that, under a worst-case formulation, both k-nearest neighbors and kernel ridge regression induce hyperspherical uniformity. More precisely, we show that uniform distributions on manifolds are optimal for k-nearest neighbors, and that the uniform distribution on the sphere is optimal for kernel ridge regression with both the exponential dot-product kernel and the linear kernel. This theoretical insight reveals a fundamental limitation of Gaussian embeddings: their non-uniform density induces anisotropic k-NN neighborhoods, severely biasing the estimator. To correct this, we introduce SPHERE-JEPA, a theoretically grounded SSL framework. We adapt LeJEPA's Cram{é}r-Wold projection mechanism to enforce hyperspherical uniformity rather than a Gaussian prior. Empirically, SPHERE-JEPA yields significant improvements, boosting texture retrieval mAP by over 6%, while consistently matching or outperforming LeJEPA on standard benchmarks-including a +1.8% linear probing gain on ImageNet-1K (ViT-B/14).
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.
Rafael Pereira Eufrazio, Eduardo Fernandes Montesuma, Charles Casimiro Cavalcante