Many machine learning problems involve data supported on curved spaces such as spheres, rotation groups, hyperbolic spaces, and general Riemannian manifolds, where Euclidean geometry can distort distances, averages, and the resulting optimal transport (OT) problem. Existing manifold OT methods have pursued amortized out-of-sample maps, while entropic regularization has made discrete OT more scalable, but these advantages have remained largely disjoint. We propose Entropic Riemannian Neural Optimal Transport (Entropic RNOT), a unified framework that combines intrinsic entropic OT with amortized out-of-sample evaluation on Riemannian manifolds. Our method learns a single target-side Schrödinger potential through a neural pullback parameterization, recovers the induced Gibbs coupling, and uses the resulting conditional laws to construct intrinsic transport surrogates. These include barycentric projections on Cartan-Hadamard manifolds and heat-smoothed conditional surrogates on stochastically complete manifolds, the latter turning possibly atomic target laws into absolutely continuous ones. For fixed regularization $\varepsilon>0$, we prove that the proposed hypothesis class recovers the entropic optimal coupling in strong probabilistic metrics. As consequences, barycentric surrogates converge in $L^2$, while heat-smoothed surrogates are stable at fixed heat time and asymptotically unbiased as the heat time vanishes. The guarantees hold for compactly supported data on possibly noncompact manifolds. Empirically, our method matches or improves over Euclidean, tangent-space, and log-Euclidean baselines on benchmarks over $\mathbb{S}^2$, $\mathrm{SO}(3)$, $\mathrm{SPD}(3)$, $\mathrm{SE}(3)$, and $\mathbb{H}^2$, scales favorably relative to discrete manifold Sinkhorn, and in a protein-ligand docking application, refines poses on $\mathrm{SE}(3)$ without retraining or per-instance optimization.

In a number of disciplines, the data (e.g., graphs, manifolds) to be analyzed are non-Euclideaninnature.

Figure 2: These two graphs cannot be distinguished by 1-WL-test. The COMBINE step takes the result of AGGREGATE and the previous representation of current node asinput. Wereduce theFFN inner-layer dimension of4din [47] tod, which does not appreciably hurt the performance but significantly save the parameters. The embedding dropout ratio is set to 0.1 by default in many previous Transformer works[11,34]. The rest of hyper-parameters remain unchanged. Table 8 summarizes the hyper-parameters used for fine-tuning Graphormer on OGBGMolPCBA.

Reinforcement learning typically relies heavily on a well-designed reward signal, which gets more challenging in cooperative multi-agent reinforcement learning. Alternatively, unsupervised reinforcement learning (URL) has delivered on its promise in the recent past to learn useful skills and explore the environment without external supervised signals. These approaches mainly aimed for the single agent to reach distinguishable states, insufficient for multi-agent systems due to that each agent interacts with not only the environment, but also the other agents. We propose Synergy Pattern Diversifying Oriented Unsupervised Multi-agent Reinforcement Learning (SPD) to learn generic coordination policies for agents with no extrinsic reward. Specifically, we devise the Synergy Pattern Graph (SPG), a graph depicting the relationships of agents at each time step. Furthermore, we propose an episode-wise divergence measurement to approximate the discrepancy of synergy patterns. To overcome the challenge of sparse return, we decompose the discrepancy of synergy patterns to per-time-step pseudo-reward. Empirically, we show the capacity of SPD to acquire meaningful coordination policies, such as maintaining specific formations in Multi-Agent Particle Environment and pass-and-shoot in Google Research Football. Furthermore, we demonstrate that the same instructive pretrained policy's parameters can serve as a good initialization for a series of downstream tasks' policies, achieving higher data efficiency and outperforming state-of-the-art approaches in Google Research Football.

Vision-Language Models (VLMs) have become indispensable for multimodal reasoning, yet their representations often encode and amplify demographic biases, resulting in biased associations and misaligned predictions in downstream tasks. Such behavior undermines fairness and distorts the intended alignment between vision and language. Recent post-hoc approaches attempt to mitigate bias by replacing the most attribute-correlated embedding coordinates with neutral values. However, our systematic analysis reveals three critical failures of this coordinate-wise approach: feature entanglement, poor cross-dataset generalization, and incomplete bias removal. We find that bias is not localized to a few coordinates but is instead distributed across a few linear subspaces. To address these limitations, we propose $\textbf{S}$ubspace $\textbf{P}$rojection $\textbf{D}$ebiasing ($\textbf{SPD}$), a geometrically principled framework that identifies and removes the entire subspace of linearly decodable bias while reinserting a neutral mean component to preserve semantic fidelity. Extensive experiments across zero-shot classification, text-to-image retrieval, and image generation validate the effectiveness of SPD: our method achieves more robust debiasing with an average improvement of $18.5\%$ across four fairness metrics, while maintaining minimal loss in task performance compared to the best debiasing baseline.

In a number of disciplines, the data (e.g., graphs, manifolds) to be analyzed are