Recent works have demonstrated promising performances of neural networks on hyperbolic spaces and symmetric positive definite (SPD) manifolds. These spaces belong to a family of Riemannian manifolds referred to as symmetric spaces of noncompact type. In this paper, we propose a novel approach for developing neural networks on such spaces. Our approach relies on a unified formulation of the distance from a point to a hyperplane on the considered spaces. We show that some existing formulations of the point-to-hyperplane distance can be recovered by our approach under specific settings. Furthermore, we derive a closed-form expression for the point-to-hyperplane distance in higher-rank symmetric spaces of noncompact type equipped with G-invariant Riemannian metrics. The derived distance then serves as a tool to design fully-connected (FC) layers and an attention mechanism for neural networks on the considered spaces. Our approach is validated on challenging benchmarks for image classification, electroencephalogram (EEG) signal classification, image generation, and natural language inference.

Large Language Models (LLMs) often struggle with generating truly innovative ideas, typically defaulting to high-probability, familiar concepts within their training data's "gravity wells." While advanced search-based methods like Tree of Thoughts (ToT) attempt to mitigate this, they are fundamentally limited by their reliance on unprincipled, inconsistent self-evaluation heuristics to guide exploration. To address this gap, we introduce \textbf{Magellan}, a novel framework that reframes creative generation as a principled, guided exploration of an LLM's latent conceptual space. At its core, Magellan employs Monte Carlo Tree Search (MCTS) governed by a hierarchical guidance system. For long-range direction, a "semantic compass" vector, formulated via orthogonal projection, steers the search towards relevant novelty. For local, step-by-step decisions, a landscape-aware value function replaces flawed self-evaluation with an explicit reward structure that balances intrinsic coherence, extrinsic novelty, and narrative progress. Extensive experiments demonstrate that Magellan significantly outperforms strong baselines, including ReAct and ToT, in generating scientific ideas with superior plausibility and innovation. Our work shows that for creative discovery, a principled, guided search is more effective than unconstrained agency, paving the way for LLMs to become more capable partners in innovation.

Recent work has demonstrated the potential of diffusion models in robot bimanual skill learning. However, existing methods ignore the learning of posture-dependent task features, which are crucial for adapting dual-arm configurations to meet specific force and velocity requirements in dexterous bimanual manipulation. To address this limitation, we propose Manipulability-Aware Diffusion Policy (ManiDP), a novel imitation learning method that not only generates plausible bimanual trajectories, but also optimizes dual-arm configurations to better satisfy posture-dependent task requirements. ManiDP achieves this by extracting bimanual manipulability from expert demonstrations and encoding the encapsulated posture features using Riemannian-based probabilistic models. These encoded posture features are then incorporated into a conditional diffusion process to guide the generation of task-compatible bimanual motion sequences. We evaluate ManiDP on six real-world bimanual tasks, where the experimental results demonstrate a 39.33$\%$ increase in average manipulation success rate and a 0.45 improvement in task compatibility compared to baseline methods. This work highlights the importance of integrating posture-relevant robotic priors into bimanual skill diffusion to enable human-like adaptability and dexterity.

Riemannian neural networks, which extend deep learning techniques to Riemannian spaces, have gained significant attention in machine learning.

Result diversification (RD) is a crucial technique in Text-to-Image Retrieval for enhancing the efficiency of a practical application. Conventional methods focus solely on increasing the diversity metric of image appearances. However, the diversity metric and its desired value vary depending on the application, which limits the applications of RD. This paper proposes a novel task called CDR-CA (Contextual Diversity Refinement of Composite Attributes). CDR-CA aims to refine the diversities of multiple attributes, according to the application's context. To address this task, we propose Multi-Source DPPs, a simple yet strong baseline that extends the Determinantal Point Process (DPP) to multi-sources. We model MS-DPP as a single DPP model with a unified similarity matrix based on a manifold representation. We also introduce Tangent Normalization to reflect contexts. Extensive experiments demonstrate the effectiveness of the proposed method.

Covariance matrices have proven highly effective across many scientific fields. Since these matrices lie within the Symmetric Positive Definite (SPD) manifold - a Riemannian space with intrinsic non-Euclidean geometry, the primary challenge in representation learning is to respect this underlying geometric structure. Drawing inspiration from the success of Euclidean deep learning, researchers have developed neural networks on the SPD manifolds for more faithful covariance embedding learning. A notable advancement in this area is the implementation of Riemannian batch normalization (RBN), which has been shown to improve the performance of SPD network models. Nonetheless, the Riemannian metric beneath the existing RBN might fail to effectively deal with the ill-conditioned SPD matrices (ICSM), undermining the effectiveness of RBN. In contrast, the Bures-Wasserstein metric (BWM) demonstrates superior performance for ill-conditioning. In addition, the recently introduced Generalized BWM (GBWM) parameterizes the vanilla BWM via an SPD matrix, allowing for a more nuanced representation of vibrant geometries of the SPD manifold. Therefore, we propose a novel RBN algorithm based on the GBW geometry, incorporating a learnable metric parameter. Moreover, the deformation of GBWM by matrix power is also introduced to further enhance the representational capacity of GBWM-based RBN. Experimental results on different datasets validate the effectiveness of our proposed method.

By incorporating physical consistency as inductive bias, deep neural networks display increased generalization capabilities and data efficiency in learning nonlinear dynamic models. However, the complexity of these models generally increases with the system dimensionality, requiring larger datasets, more complex deep networks, and significant computational effort. We propose a novel geometric network architecture to learn physically-consistent reduced-order dynamic parameters that accurately describe the original high-dimensional system behavior. This is achieved by building on recent advances in model-order reduction and by adopting a Riemannian perspective to jointly learn a non-linear structure-preserving latent space and the associated low-dimensional dynamics. Our approach enables accurate long-term predictions of the high-dimensional dynamics of rigid and deformable systems with increased data efficiency by inferring interpretable and physically plausible reduced Lagrangian models.