Forecasting Continuous Non-Conservative Dynamical Systems in SO(3)

Modeling the rotation of moving objects is a fundamental task in computer vision, yet $SO(3)$ extrapolation still presents numerous challenges: (1) unknown quantities such as the moment of inertia complicate dynamics, (2) the presence of external forces and torques can lead to non-conservative kinematics, and (3) estimating evolving state trajectories under sparse, noisy observations requires robustness. We propose modeling trajectories of noisy pose estimates on the manifold of 3D rotations in a physically and geometrically meaningful way by leveraging Neural Controlled Differential Equations guided with $SO(3)$ Savitzky-Golay paths. Existing extrapolation methods often rely on energy conservation or constant velocity assumptions, limiting their applicability in real-world scenarios involving non-conservative forces. In contrast, our approach is agnostic to energy and momentum conservation while being robust to input noise, making it applicable to complex, non-inertial systems. Our approach is easily integrated as a module in existing pipelines and generalizes well to trajectories with unknown physical parameters. By learning to approximate object dynamics from noisy states during training, our model attains robust extrapolation capabilities in simulation and various real-world settings. Code is available at https://github.com/bastianlb/forecasting-rotational-dynamics

On the Interaction of Compressibility and Adversarial Robustness

Modern neural networks are expected to simultaneously satisfy a host of desirable properties: accurate fitting to training data, generalization to unseen inputs, parameter and computational efficiency, and robustness to adversarial perturbations. While compressibility and robustness have each been studied extensively, a unified understanding of their interaction still remains elusive. In this work, we develop a principled framework to analyze how different forms of compressibility - such as neuron-level sparsity and spectral compressibility - affect adversarial robustness. We show that these forms of compression can induce a small number of highly sensitive directions in the representation space, which adversaries can exploit to construct effective perturbations. Our analysis yields a simple yet instructive robustness bound, revealing how neuron and spectral compressibility impact $L_\infty$ and $L_2$ robustness via their effects on the learned representations. Crucially, the vulnerabilities we identify arise irrespective of how compression is achieved - whether via regularization, architectural bias, or implicit learning dynamics. Through empirical evaluations across synthetic and realistic tasks, we confirm our theoretical predictions, and further demonstrate that these vulnerabilities persist under adversarial training and transfer learning, and contribute to the emergence of universal adversarial perturbations. Our findings show a fundamental tension between structured compressibility and robustness, and suggest new pathways for designing models that are both efficient and secure.

Large Learning Rates Simultaneously Achieve Robustness to Spurious Correlations and Compressibility

Robustness and resource-efficiency are two highly desirable properties for modern machine learning models. However, achieving them jointly remains a challenge. In this paper, we position high learning rates as a facilitator for simultaneously achieving robustness to spurious correlations and network compressibility. We demonstrate that large learning rates also produce desirable representation properties such as invariant feature utilization, class separation, and activation sparsity. Importantly, our findings indicate that large learning rates compare favorably to other hyperparameters and regularization methods, in consistently satisfying these properties in tandem. In addition to demonstrating the positive effect of large learning rates across diverse spurious correlation datasets, models, and optimizers, we also present strong evidence that the previously documented success of large learning rates in standard classification tasks is likely due to its effect on addressing hidden/rare spurious correlations in the training dataset.

Mutual Information Free Topological Generalization Bounds via Stability

Providing generalization guarantees for stochastic optimization algorithms is a major challenge in modern learning theory. Recently, several studies highlighted the impact of the geometry of training trajectories on the generalization error, both theoretically and empirically. Among these works, a series of topological generalization bounds have been proposed, relating the generalization error to notions of topological complexity that stem from topological data analysis (TDA). Despite their empirical success, these bounds rely on intricate information-theoretic (IT) terms that can be bounded in specific cases but remain intractable for practical algorithms (such as ADAM), potentially reducing the relevance of the derived bounds. In this paper, we seek to formulate comprehensive and interpretable topological generalization bounds free of intractable mutual information terms. To this end, we introduce a novel learning theoretic framework that departs from the existing strategies via proof techniques rooted in algorithmic stability. By extending an existing notion of \textit{hypothesis set stability}, to \textit{trajectory stability}, we prove that the generalization error of trajectory-stable algorithms can be upper bounded in terms of (i) TDA quantities describing the complexity of the trajectory of the optimizer in the parameter space, and (ii) the trajectory stability parameter of the algorithm. Through a series of experimental evaluations, we demonstrate that the TDA terms in the bound are of great importance, especially as the number of training samples grows. This ultimately forms an explanation of the empirical success of the topological generalization bounds.

Continuous-Time SO(3) Forecasting with Savitzky--Golay Neural Controlled Differential Equations

Tracking and forecasting the rotation of objects is fundamental in computer vision and robotics, yet SO(3) extrapolation remains challenging as (1) sensor observations can be noisy and sparse, (2) motion patterns can be governed by complex dynamics, and (3) application settings can demand long-term forecasting. This work proposes modeling continuous-time rotational object dynamics on $SO(3)$ using Neural Controlled Differential Equations guided by Savitzky-Golay paths. Unlike existing methods that rely on simplified motion assumptions, our method learns a general latent dynamical system of the underlying object trajectory while respecting the geometric structure of rotations. Experimental results on real-world data demonstrate compelling forecasting capabilities compared to existing approaches.

Copresheaf Topological Neural Networks: A Generalized Deep Learning Framework

We introduce copresheaf topological neural networks (CTNNs), a powerful and unifying framework that encapsulates a wide spectrum of deep learning architectures, designed to operate on structured data: including images, point clouds, graphs, meshes, and topological manifolds. While deep learning has profoundly impacted domains ranging from digital assistants to autonomous systems, the principled design of neural architectures tailored to specific tasks and data types remains one of the field's most persistent open challenges. CTNNs address this gap by grounding model design in the language of copresheaves, a concept from algebraic topology that generalizes and subsumes most practical deep learning models in use today. This abstract yet constructive formulation yields a rich design space from which theoretically sound and practically effective solutions can be derived to tackle core challenges in representation learning: long-range dependencies, oversmoothing, heterophily, and non-Euclidean domains. Our empirical results on structured data benchmarks demonstrate that CTNNs consistently outperform conventional baselines, particularly in tasks requiring hierarchical or localized sensitivity. These results underscore CTNNs as a principled, multi-scale foundation for the next generation of deep learning architectures.

Towards Dynamic 3D Reconstruction of Hand-Instrument Interaction in Ophthalmic Surgery

Accurate 3D reconstruction of hands and instruments is critical for vision-based analysis of ophthalmic microsurgery, yet progress has been hampered by the lack of realistic, large-scale datasets and reliable annotation tools. In this work, we introduce OphNet-3D, the first extensive RGB-D dynamic 3D reconstruction dataset for ophthalmic surgery, comprising 41 sequences from 40 surgeons and totaling 7.1 million frames, with fine-grained annotations of 12 surgical phases, 10 instrument categories, dense MANO hand meshes, and full 6-DoF instrument poses. To scalably produce high-fidelity labels, we design a multi-stage automatic annotation pipeline that integrates multi-view data observation, data-driven motion prior with cross-view geometric consistency and biomechanical constraints, along with a combination of collision-aware interaction constraints for instrument interactions. Building upon OphNet-3D, we establish two challenging benchmarks-bimanual hand pose estimation and hand-instrument interaction reconstruction-and propose two dedicated architectures: H-Net for dual-hand mesh recovery and OH-Net for joint reconstruction of two-hand-two-instrument interactions. These models leverage a novel spatial reasoning module with weak-perspective camera modeling and collision-aware center-based representation. Both architectures outperform existing methods by substantial margins, achieving improvements of over 2mm in Mean Per Joint Position Error (MPJPE) and up to 23% in ADD-S metrics for hand and instrument reconstruction, respectively.

Efficient Training of Neural SDEs Using Stochastic Optimal Control

We present a hierarchical, control theory inspired method for variational inference (VI) for neural stochastic differential equations (SDEs). While VI for neural SDEs is a promising avenue for uncertainty-aware reasoning in time-series, it is computationally challenging due to the iterative nature of maximizing the ELBO. In this work, we propose to decompose the control term into linear and residual non-linear components and derive an optimal control term for linear SDEs, using stochastic optimal control. Modeling the non-linear component by a neural network, we show how to efficiently train neural SDEs without sacrificing their expressive power. Since the linear part of the control term is optimal and does not need to be learned, the training is initialized at a lower cost and we observe faster convergence.

A Flag Decomposition for Hierarchical Datasets

Flag manifolds encode hierarchical nested sequences of subspaces and serve as powerful structures for various computer vision and machine learning applications. Despite their utility in tasks such as dimensionality reduction, motion averaging, and subspace clustering, current applications are often restricted to extracting flags using common matrix decomposition methods like the singular value decomposition. Here, we address the need for a general algorithm to factorize and work with hierarchical datasets. In particular, we propose a novel, flag-based method that decomposes arbitrary hierarchical real-valued data into a hierarchy-preserving flag representation in Stiefel coordinates. Our work harnesses the potential of flag manifolds in applications including denoising, clustering, and few-shot learning.

HOG-Diff: Higher-Order Guided Diffusion for Graph Generation

Graph generation is a critical yet challenging task as empirical analyses require a deep understanding of complex, non-Euclidean structures. Although diffusion models have recently made significant achievements in graph generation, these models typically adapt from the frameworks designed for image generation, making them ill-suited for capturing the topological properties of graphs. In this work, we propose a novel Higher-order Guided Diffusion (HOG-Diff) model that follows a coarse-to-fine generation curriculum and is guided by higher-order information, enabling the progressive generation of plausible graphs with inherent topological structures. We further prove that our model exhibits a stronger theoretical guarantee than classical diffusion frameworks. Extensive experiments on both molecular and generic graph generation tasks demonstrate that our method consistently outperforms or remains competitive with state-of-the-art baselines. Our code is available at https://github.com/Yiminghh/HOG-Diff.