Capsule Networks (CapsNets) show exceptional graph representation capacity via dynamic routing and vectorized hierarchical representations, but they model the complex geometries of real\-world graphs poorly by fixed\-curvature space due to the inherent geodesical disconnectedness issues, leading to suboptimal performance. Recent works find that non\-Euclidean pseudo\-Riemannian manifolds provide specific inductive biases for embedding graph data, but how to leverage them to improve CapsNets is still underexplored. Here, we extend the Euclidean capsule routing into geodesically disconnected pseudo\-Riemannian manifolds and derive a Pseudo\-Riemannian Capsule Network (PR\-CapsNet), which models data in pseudo\-Riemannian manifolds of adaptive curvature, for graph representation learning. Specifically, PR\-CapsNet enhances the CapsNet with Adaptive Pseudo\-Riemannian Tangent Space Routing by utilizing pseudo\-Riemannian geometry. Unlike single\-curvature or subspace\-partitioning methods, PR\-CapsNet concurrently models hierarchical and cluster or cyclic graph structures via its versatile pseudo\-Riemannian metric. It first deploys Pseudo\-Riemannian Tangent Space Routing to decompose capsule states into spherical\-temporal and Euclidean\-spatial subspaces with diffeomorphic transformations. Then, an Adaptive Curvature Routing is developed to adaptively fuse features from different curvature spaces for complex graphs via a learnable curvature tensor with geometric attention from local manifold properties. Finally, a geometric properties\-preserved Pseudo\-Riemannian Capsule Classifier is developed to project capsule embeddings to tangent spaces and use curvature\-weighted softmax for classification. Extensive experiments on node and graph classification benchmarks show PR\-CapsNet outperforms SOTA models, validating PR\-CapsNet's strong representation power for complex graph structures.

Analysing how neural networks represent data features in their activations can help interpret how they perform tasks. Hence, a long line of work has focused on mathematically characterising the geometry of such "neural representations." In parallel, machine learning has seen a surge of interest in understanding how dynamical systems perform computations on time-varying input data. Yet, the link between computation-through-dynamics and representational geometry remains poorly understood. Here, we hypothesise that recurrent neural networks (RNNs) perform computations by dynamically warping their representations of task variables. To test this hypothesis, we develop a Riemannian geometric framework that enables the derivation of the manifold topology and geometry of a dynamical system from the manifold of its inputs. By characterising the time-varying geometry of RNNs, we show that dynamic warping is a fundamental feature of their computations.

Robust decoding and classification of brain patterns measured with electroencephalography (EEG) remains a major challenge for real-world (i.e. outside scientific lab and medical facilities) brain-computer interface (BCI) applications due to well documented inter- and intra-participant variability. Here, we present a large-scale benchmark evaluating over 340,000+ unique combinations of spatial and nonlinear EEG classification. Our methodological pipeline consists in combinations of Common Spatial Patterns (CSP), Riemannian geometry, functional connectivity, and fractal- or entropy-based features across three open-access EEG datasets. Unlike prior studies, our analysis operates at the per-participant level and across multiple frequency bands (8-15 Hz and 8-30 Hz), enabling direct assessment of both group-level performance and individual variability. Covariance tangent space projection (cov-tgsp) and CSP consistently achieved the highest average classification accuracies. However, their effectiveness was strongly dataset-dependent, and marked participant-level differences persisted, particularly in the most heterogeneous of the datasets. Importantly, nonlinear methods outperformed spatial approaches for specific individuals, underscoring the need for personalized pipeline selection. Our findings highlight that no universal 'one-size-fits-all' method can optimally decode EEG motor imagery patterns across all users or datasets. Future work will require adaptive, multimodal, and possibly novel approaches to fully address neurophysiological variability in practical BCI applications where the system can automatically adapt to what makes each user unique.

Automated personality and soft skill assessment from multimodal behavioral data remains challenging due to limited datasets and methods that fail to capture geometric structure inherent in human traits. We introduce RecruitView, a dataset of 2,011 naturalistic video interview clips from 300+ participants with 27,000 pairwise comparative judgments across 12 dimensions: Big Five personality traits, overall personality score, and six interview performance metrics. To leverage this data, we propose Cross-Modal Regression with Manifold Fusion (CRMF), a geometric deep learning framework that explicitly models behavioral representations across hyperbolic, spherical, and Euclidean manifolds. CRMF employs geometry-specific expert networks to capture hierarchical trait structures, directional behavioral patterns, and continuous performance variations simultaneously. An adaptive routing mechanism dynamically weights expert contributions based on input characteristics. Through principled tangent space fusion, CRMF achieves superior performance while training 40-50% fewer trainable parameters than large multimodal models. Extensive experiments demonstrate that CRMF substantially outperforms the selected baselines, achieving up to 11.4% improvement in Spearman correlation and 6.0% in concordance index. Our RecruitView dataset is publicly available at https://huggingface.co/datasets/AI4A-lab/RecruitView

Adversarial examples in neural networks have been extensively studied in Euclidean geometry, but recent advances in \textit{hyperbolic networks} call for a reevaluation of attack strategies in non-Euclidean geometries. Existing methods such as FGSM and PGD apply perturbations without regard to the underlying hyperbolic structure, potentially leading to inefficient or geometrically inconsistent attacks. In this work, we propose a novel adversarial attack that explicitly leverages the geometric properties of hyperbolic space. Specifically, we compute the gradient of the loss function in the tangent space of hyperbolic space, decompose it into a radial (depth) component and an angular (semantic) component, and apply perturbation derived solely from the angular direction. Our method generates adversarial examples by focusing perturbations in semantically sensitive directions encoded in angular movement within the hyperbolic geometry. Empirical results on image classification, cross-modal retrieval tasks and network architectures demonstrate that our attack achieves higher fooling rates than conventional adversarial attacks, while producing high-impact perturbations with deeper insights into vulnerabilities of hyperbolic embeddings. This work highlights the importance of geometry-aware adversarial strategies in curved representation spaces and provides a principled framework for attacking hierarchical embeddings.

Cross-subject motor-imagery decoding remains a major challenge in EEG-based brain-computer interfaces due to strong subject variability and the curved geometry of covariance matrices on the symmetric positive definite (SPD) manifold. We address the zero-shot cross-subject setting, where no target-subject labels or adaptation are allowed, by introducing novel geometry-aware preprocessing modules and deep congruence networks that operate directly on SPD covariance matrices. Our preprocessing modules, DCR and RiFU, extend Riemannian Alignment by improving action separation while reducing subject-specific distortions. We further propose two manifold classifiers, SPD-DCNet and RiFUNet, which use hierarchical congruence transforms to learn discriminative, subject-invariant covariance representations. On the BCI-IV 2a benchmark, our framework improves cross-subject accuracy by 3-4% over the strongest classical baselines, demonstrating the value of geometry-aware transformations for robust EEG decoding.

In most applications of interest (e.g., modeling images), we have