Analysis of learned representations has a blind spot: it focuses on $similarity$, measuring how closely embeddings align with external references, but similarity reveals only what is represented, not whether that structure is robust. We introduce $geometric$ $stability$, a distinct dimension that quantifies how reliably representational geometry holds under perturbation, and present $Shesha$, a framework for measuring it. Across 2,463 configurations in seven domains, we show that stability and similarity are empirically uncorrelated ($ρ\approx 0.01$) and mechanistically distinct: similarity metrics collapse after removing the top principal components, while stability retains sensitivity to fine-grained manifold structure. This distinction yields actionable insights: for safety monitoring, stability acts as a functional geometric canary, detecting structural drift nearly 2$\times$ more sensitively than CKA while filtering out the non-functional noise that triggers false alarms in rigid distance metrics; for controllability, supervised stability predicts linear steerability ($ρ= 0.89$-$0.96$); for model selection, stability dissociates from transferability, revealing a geometric tax that transfer optimization incurs. Beyond machine learning, stability predicts CRISPR perturbation coherence and neural-behavioral coupling. By quantifying $how$ $reliably$ systems maintain structure, geometric stability provides a necessary complement to similarity for auditing representations across biological and computational systems.

Physics-Informed Neural Networks (PINNs) have emerged as a powerful tool for solving differential equations and modeling physical systems by embedding physical laws into the learning process. However, rigorously quantifying how well a PINN captures the complete dynamical behavior of the system, beyond simple trajectory prediction, remains a challenge. This paper proposes a novel experimental framework to address this by employing Fisher information for differentiable dynamical systems, denoted $g_F^C$. This Fisher information, distinct from its statistical counterpart, measures inherent uncertainties in deterministic systems, such as sensitivity to initial conditions, and is related to the phase space curvature and the net stretching action of the state space evolution. We hypothesize that if a PINN accurately learns the underlying dynamics of a physical system, then the Fisher information landscape derived from the PINN's learned equations of motion will closely match that of the original analytical model. This match would signify that the PINN has achieved comprehensive fidelity capturing not only the state evolution but also crucial geometric and stability properties. We outline an experimental methodology using the dynamical model of a car to compute and compare $g_F^C$ for both the analytical model and a trained PINN. The comparison, based on the Jacobians of the respective system dynamics, provides a quantitative measure of the PINN's fidelity in representing the system's intricate dynamical characteristics.

LiDAR point cloud semantic segmentation enables the robots to obtain fine-grained semantic information of the surrounding environment. Recently, many works project the point cloud onto the 2D image and adopt the 2D Convolutional Neural Networks (CNNs) or vision transformer for LiDAR point cloud semantic segmentation. However, since more than one point can be projected onto the same 2D position but only one point can be preserved, the previous 2D projection-based segmentation methods suffer from inevitable quantized information loss, which results in incomplete geometric structure, especially for small objects. To avoid quantized information loss, in this paper, we propose a novel spherical frustum structure, which preserves all points projected onto the same 2D position. Additionally, a hash-based representation is proposed for memory-efficient spherical frustum storage. Based on the spherical frustum structure, the Spherical Frustum sparse Convolution (SFC) and Frustum Farthest Point Sampling (F2PS) are proposed to convolve and sample the points stored in spherical frustums respectively. Finally, we present the Spherical Frustum sparse Convolution Network (SFCNet) to adopt 2D CNNs for LiDAR point cloud semantic segmentation without quantized information loss. Extensive experiments on the SemanticKITTI and nuScenes datasets demonstrate that our SFCNet outperforms previous 2D projection-based semantic segmentation methods based on conventional spherical projection and shows better performance on small object segmentation by preserving complete geometric structure. Codes will be available at https://github.com/IRMVLab/SFCNet.

Since Nesterov's seminal 1983 work, many accelerated first-order optimization methods have been proposed, but their analyses lacks a common unifying structure. In this work, we identify a geometric structure satisfied by a wide range of first-order accelerated methods. Using this geometric insight, we present several novel generalizations of accelerated methods. Most interesting among them is a method that reduces the squared gradient norm with $\mathcal{O}(1/K^4)$ rate in the prox-grad setup, faster than the $\mathcal{O}(1/K^3)$ rates of Nesterov's FGM or Kim and Fessler's FPGM-m.

We propose geodesic-based optimization methods on dually flat spaces, where the geometric structure of the parameter manifold is closely related to the form of the objective function. A primary application is maximum likelihood estimation in statistical models, especially exponential families, whose model manifolds are dually flat. We show that an m-geodesic update, which directly optimizes the log-likelihood, can theoretically reach the maximum likelihood estimator in a single step. In contrast, an e-geodesic update has a practical advantage in cases where the parameter space is geodesically complete, allowing optimization without explicitly handling parameter constraints. We establish the theoretical properties of the proposed methods and validate their effectiveness through numerical experiments.

The non-commutative nature of 3D rotations poses well-known challenges in generalizing planar problems to three-dimensional ones, even more so in contact-rich tasks where haptic information (i.e., forces/torques) is involved. In this sense, not all learning-based algorithms that are currently available generalize to 3D orientation estimation. Non-linear filters defined on $\mathbf{\mathbb{SO}(3)}$ are widely used with inertial measurement sensors; however, none of them have been used with haptic measurements. This paper presents a unique complementary filtering framework that interprets the geometric shape of objects in the form of superquadrics, exploits the symmetry of $\mathbf{\mathbb{SO}(3)}$, and uses force and vision sensors as measurements to provide an estimate of orientation. The framework's robustness and almost global stability are substantiated by a set of experiments on a dual-arm robotic setup.

Pretrained transformers provide rich, general-purpose embeddings, which are transferred to downstream tasks. However, current transfer strategies: fine-tuning and probing, either distort the pretrained geometric structure of the embeddings or lack sufficient expressivity to capture task-relevant signals. These issues become even more pronounced when supervised data are scarce. Here, we introduce Freeze, Diffuse, Decode (FDD), a novel diffusion-based framework that adapts pre-trained embeddings to downstream tasks while preserving their underlying geometric structure. FDD propagates supervised signal along the intrinsic manifold of frozen embeddings, enabling a geometry-aware adaptation of the embedding space. Applied to antimicrobial peptide design, FDD yields low-dimensional, predictive, and interpretable representations that support property prediction, retrieval, and latent-space interpolation.

Existing EEG foundation models mainly treat neural signals as generic time series in Euclidean space, ignoring the intrinsic geometric structure of neural dynamics that constrains brain activity to low-dimensional manifolds. This fundamental mismatch between model assumptions and neural geometry limits representation quality and cross-subject generalization. ManifoldFormer addresses this limitation through a novel geometric deep learning framework that explicitly learns neural manifold representations. The architecture integrates three key innovations: a Riemannian VAE for manifold embedding that preserves geometric structure, a geometric Transformer with geodesic-aware attention mechanisms operating directly on neural manifolds, and a dynamics predictor leveraging neural ODEs for manifold-constrained temporal evolution. Extensive evaluation across four public datasets demonstrates substantial improvements over state-of-the-art methods, with 4.6-4.8% higher accuracy and 6.2-10.2% higher Cohen's Kappa, while maintaining robust cross-subject generalization. The geometric approach reveals meaningful neural patterns consistent with neurophysiological principles, establishing geometric constraints as essential for effective EEG foundation models.

Artificial intelligence systems in critical fields like autonomous driving and medical imaging analysis often continually learn new tasks using a shared stream of input data. For instance, after learning to detect traffic signs, a model may later need to learn to classify traffic lights or different types of vehicles using the same camera feed. This scenario introduces a challenging setting we term Continual Multitask Learning (CMTL), where a model sequentially learns new tasks on an underlying data distribution without forgetting previously learned abilities. Existing continual learning methods often fail in this setting because they learn fragmented, task-specific features that interfere with one another. To address this, we introduce Learning with Preserving (LwP), a novel framework that shifts the focus from preserving task outputs to maintaining the geometric structure of the shared representation space. The core of LwP is a Dynamically Weighted Distance Preservation (DWDP) loss that prevents representation drift by regularizing the pairwise distances between latent data representations. This mechanism of preserving the underlying geometric structure allows the model to retain implicit knowledge and support diverse tasks without requiring a replay buffer, making it suitable for privacy-conscious applications. Extensive evaluations on time-series and image benchmarks show that LwP not only mitigates catastrophic forgetting but also consistently outperforms state-of-the-art baselines in CMTL tasks. Notably, our method shows superior robustness to distribution shifts and is the only approach to surpass the strong single-task learning baseline, underscoring its effectiveness for real-world dynamic environments.

Grassmannian manifold offers a powerful carrier for geometric representation learning by modelling high-dimensional data as low-dimensional subspaces. However, existing approaches predominantly rely on static single-subspace representations, neglecting the dynamic interplay between multiple subspaces critical for capturing complex geometric structures. To address this limitation, we propose a topology-driven multi-subspace fusion framework that enables adaptive subspace collaboration on the Grassmannian. Our solution introduces two key innovations: (1) Inspired by the Kolmogorov-Arnold representation theorem, an adaptive multi-subspace modelling mechanism is proposed that dynamically selects and weights task-relevant subspaces via topological convergence analysis, and (2) a multi-subspace interaction block that fuses heterogeneous geometric representations through Fr echet mean optimisation on the manifold. Theoretically, we establish the convergence guarantees of adaptive subspaces under a projection metric topology, ensuring stable gradient-based optimisation. Practically, we integrate Riemannian batch normalisation and mutual information regularisation to enhance discriminability and robustness. Extensive experiments on 3D action recognition (HDM05, FPHA), EEG classification (MAMEM-SSVEPII), and graph tasks demonstrate state-of-the-art performance. Our work not only advances geometric deep learning but also successfully adapts the proven multi-channel interaction philosophy of Euclidean networks to non-Euclidean domains, achieving superior discriminability and interpretability.