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Numerous applications in machine learning and data analytics can be formulated as equilibrium computation over Riemannian manifolds. Despite the extensive investigation of their Euclidean counterparts, the performance of Riemannian gradient-based algorithms remain opaque and poorly understood. We revisit the original scheme of Riemannian gradient descent (RGD) and analyze it under a geodesic monotonicity assumption, which includes the well-studied geodesically convex-concave min-max optimization problem as a special case. Our main contribution is to show that, despite the phenomenon of distance distortion, the RGD scheme, with a step size that is agnostic to the manifold's curvature, achieves a curvature-independent and linear last-iterate convergence rate in the geodesically strongly monotone setting. To the best of our knowledge, the possibility of curvature-independent rates and/or last-iterate convergence in the Riemannian setting has not been considered before. 

Badminton is a fast-paced sport that requires a strategic combination of spatial, temporal, and technical tactics. To gain a competitive edge at high-level competitions, badminton professionals frequently analyze match videos to gain insights and develop game strategies. However, the current process for analyzing matches is time-consuming and relies heavily on manual note-taking, due to the lack of automatic data collection and appropriate visualization tools. As a result, there is a gap in effectively analyzing matches and communicating insights among badminton coaches and players. This work proposes an end-to-end immersive match analysis pipeline designed in close collaboration with badminton professionals, including Olympic and national coaches and players. We present VIRD, a VR Bird (i.e., shuttle) immersive analysis tool, that supports interactive badminton game analysis in an immersive environment based on 3D reconstructed game views of the match video. We propose a top-down analytic workflow that allows users to seamlessly move from a high-level match overview to a detailed game view of individual rallies and shots, using situated 3D visualizations and video. We collect 3D spatial and dynamic shot data and player poses with computer vision models and visualize them in VR. Through immersive visualizations, coaches can interactively analyze situated spatial data (player positions, poses, and shot trajectories) with flexible viewpoints while navigating between shots and rallies effectively with embodied interaction. We evaluated the usefulness of VIRD with Olympic and national-level coaches and players in real matches. Results show that immersive analytics supports effective badminton match analysis with reduced context-switching costs and enhances spatial understanding with a high sense of presence. 

Labels are widely used in augmented reality (AR) to display digital information. Ensuring the readability of AR labels requires placing them occlusion-free while keeping visual linkings legible, especially when multiple labels exist in the scene. Although existing optimization-based methods, such as force-based methods, are effective in managing AR labels in static scenarios, they often struggle in dynamic scenarios with constantly moving objects. This is due to their focus on generating layouts optimal for the current moment, neglecting future moments and leading to sub-optimal or unstable layouts over time. In this work, we present RL-LABEL, a deep reinforcement learning-based method for managing the placement of AR labels in scenarios involving moving objects. RL-LABEL considers the current and predicted future states of objects and labels, such as positions and velocities, as well as the user’s viewpoint, to make informed decisions about label placement. It balances the trade-offs between immediate and long-term objectives. Our experiments on two real-world datasets show that RL-LABEL effectively learns the decision-making process for long-term optimization, outperforming two baselines (i.e., no view management and a force-based method) by minimizing label occlusions, line intersections, and label movement distance. Additionally, a user study involving 18 participants indicates that RL-LABEL excels over the baselines in aiding users to identify, compare, and summarize data on AR labels within dynamic scenes. 

We explore flow of a completely wetting fluid in a funnel, with particular focus on contact line instabilities at the fluid front. While the flow in a funnel may be related to a number of other flow configurations as limiting cases, understanding its stability is complicated due to the presence of additional azimuthal curvature, as well as due to convergent flow effects imposed by the geometry. The convergent nature of the flow leads to thickening of the film, therefore influencing its stability properties. In this work, we analyse these stability properties by combining physical experiments, asymptotic modelling, self-similar type of analysis and numerical simulations. We show that an appropriate long-wave-based model, supported by the input from experiments, simulations and linear stability analysis that originates from the flow down an incline plane, provides a basic insight allowing an understanding of the development of contact line instability and emerging length scales.