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.

Grokking at the Edge of Numerical Stability

Grokking, the sudden generalization that occurs after prolonged overfitting, is a surprising phenomenon challenging our understanding of deep learning. Although significant progress has been made in understanding grokking, the reasons behind the delayed generalization and its dependence on regularization remain unclear. In this work, we argue that without regularization, grokking tasks push models to the edge of numerical stability, introducing floating point errors in the Softmax function, which we refer to as Softmax Collapse (SC). We demonstrate that SC prevents grokking and that mitigating SC enables grokking without regularization. Investigating the root cause of SC, we find that beyond the point of overfitting, the gradients strongly align with what we call the na\"ive loss minimization (NLM) direction. This component of the gradient does not alter the model's predictions but decreases the loss by scaling the logits, typically by scaling the weights along their current direction. We show that this scaling of the logits explains the delay in generalization characteristic of grokking and eventually leads to SC, halting further learning. To validate our hypotheses, we introduce two key contributions that address the challenges in grokking tasks: StableMax, a new activation function that prevents SC and enables grokking without regularization, and $\perp$Grad, a training algorithm that promotes quick generalization in grokking tasks by preventing NLM altogether. These contributions provide new insights into grokking, elucidating its delayed generalization, reliance on regularization, and the effectiveness of existing grokking-inducing methods. Code for this paper is available at https://github.com/LucasPrietoAl/grokking-at-the-edge-of-numerical-stability.

Dyn-HaMR: Recovering 4D Interacting Hand Motion from a Dynamic Camera

We propose Dyn-HaMR, to the best of our knowledge, the first approach to reconstruct 4D global hand motion from monocular videos recorded by dynamic cameras in the wild. Reconstructing accurate 3D hand meshes from monocular videos is a crucial task for understanding human behaviour, with significant applications in augmented and virtual reality (AR/VR). However, existing methods for monocular hand reconstruction typically rely on a weak perspective camera model, which simulates hand motion within a limited camera frustum. As a result, these approaches struggle to recover the full 3D global trajectory and often produce noisy or incorrect depth estimations, particularly when the video is captured by dynamic or moving cameras, which is common in egocentric scenarios. Our Dyn-HaMR consists of a multi-stage, multi-objective optimization pipeline, that factors in (i) simultaneous localization and mapping (SLAM) to robustly estimate relative camera motion, (ii) an interacting-hand prior for generative infilling and to refine the interaction dynamics, ensuring plausible recovery under (self-)occlusions, and (iii) hierarchical initialization through a combination of state-of-the-art hand tracking methods. Through extensive evaluations on both in-the-wild and indoor datasets, we show that our approach significantly outperforms state-of-the-art methods in terms of 4D global mesh recovery. This establishes a new benchmark for hand motion reconstruction from monocular video with moving cameras. Our project page is at https://dyn-hamr.github.io/.

Convex Formulations for Training Two-Layer ReLU Neural Networks

Solving non-convex, NP-hard optimization problems is crucial for training machine learning models, including neural networks. However, non-convexity often leads to black-box machine learning models with unclear inner workings. While convex formulations have been used for verifying neural network robustness, their application to training neural networks remains less explored. In response to this challenge, we reformulate the problem of training infinite-width two-layer ReLU networks as a convex completely positive program in a finite-dimensional (lifted) space. Despite the convexity, solving this problem remains NP-hard due to the complete positivity constraint. To overcome this challenge, we introduce a semidefinite relaxation that can be solved in polynomial time. We then experimentally evaluate the tightness of this relaxation, demonstrating its competitive performance in test accuracy across a range of classification tasks.

Alignist: CAD-Informed Orientation Distribution Estimation by Fusing Shape and Correspondences

Object pose distribution estimation is crucial in robotics for better path planning and handling of symmetric objects. Recent distribution estimation approaches employ contrastive learning-based approaches by maximizing the likelihood of a single pose estimate in the absence of a CAD model. We propose a pose distribution estimation method leveraging symmetry respecting correspondence distributions and shape information obtained using a CAD model. Contrastive learning-based approaches require an exhaustive amount of training images from different viewpoints to learn the distribution properly, which is not possible in realistic scenarios. Instead, we propose a pipeline that can leverage correspondence distributions and shape information from the CAD model, which are later used to learn pose distributions. Besides, having access to pose distribution based on correspondences before learning pose distributions conditioned on images, can help formulate the loss between distributions. The prior knowledge of distribution also helps the network to focus on getting sharper modes instead. With the CAD prior, our approach converges much faster and learns distribution better by focusing on learning sharper distribution near all the valid modes, unlike contrastive approaches, which focus on a single mode at a time. We achieve benchmark results on SYMSOL-I and T-Less datasets.