Long-horizon consistency is essential for video world models, yet it becomes difficult to maintain once observed content falls outside the context window. Existing memory mechanisms either retrieve history as implicit context, which remains constrained by context capacity and entangles static and dynamic information or consolidate history into explicit global representations, which struggle to model evolving dynamics and accumulate geometric errors. To overcome these limitations, we introduce \textbf{GROME}, a novel factorized memory framework that augments video world models for long-horizon coherence across both static scenes and dynamic entities. GROME operates via three synergistic components: 1)~\emph{Static-Dynamic Factorization} decouples historical evidence into static and dynamic components, preserving static scene geometry and dynamic entity states separately; 2)~\emph{Local Memory Retrieval} selects spatially aligned static history to eliminate geometric drift alongside the latest dynamic entity states to ensure temporal continuity; 3)~\emph{Training-Free Memory Injection} seamlessly grounds the factorized memory into video world models via their native conditioning pathways. Experiments on long-horizon revisit scenarios demonstrate that GROME substantially enhances scene persistence and dynamic entity fidelity over existing methods, all while preserving the visual quality and controllability of the frozen backbone without additional training. Beyond self-rollouts, GROME readily accommodates external observations as plug-and-play memory, unlocking flexible and controllable video continuation and scene exploration.
We present GPU-accelerated Bregman Douglas--Rachford splitting algorithm (BDRS) for discrete optimal transport problem in three input formats: an explicit cost matrix, a point cloud with a ground cost between them, and a separable cost on a regular grid. For each input format, we propose hardware-aware designs of mathematically equivalent representations for the BDRS iterations to enhance numerical stability and empirical runtime. We benchmark the three proposed implementations against eight GPU baseline solvers from the literature on the same device. We demonstrate that our implementations of BDRS achieve state-of-the-art performance on their respective input formats. To the best of our knowledge, this is the first cross-solver study of GPU DOT solvers with a unified measure of optimality.
In stochastic multi-objective bandits, Pareto optimal arms may represent different trade-offs among objectives. Balanced sampling across them is important in practical settings such as online advertising. Yet, there seems to be little consensus on how to measure fairness, to our best knowledge. Achieving fairness is also tough as the set of Pareto optimal arms is unknown, and early sampling decisions may create imbalances that are difficult to erase afterwards. We measure unfairness by the KL divergence from the uniform distribution over the true Pareto set to the empirical measure induced by the sampling trajectory. Nicely, this measure admits an exact decomposition that separates the cost of sampling suboptimal arms from the imbalance among Pareto optimal arms. With a recursive form for unfairness, we develop \emph{adaptive top-two sampling}, which combines sampling counts and empirical evidence of Pareto membership to define a gain for each arm and guide sampling without committing to an estimated Pareto set. When the Pareto set is stable under small perturbations, we establish an $O(\log T/T)$ and $O(T^{-1})$ bound on expected unfairness and variance, respectively. Information-theoretic lower bounds show that the $\log T/T$ rate can be unavoidable for uniformly fair policies on stable instances, matching our upper bound. More surprisingly, stability is indeed crucial, as the expected unfairness can be $ω(\log T/T)$ without it. Finally, we create a variant that encourages exploration of undersampled arms to reduce the risk of persistent underallocation. At sufficiently large horizons, it preserves the rate on the expected unfairness while admitting a sharper bound of $O(\log^2 T/T^2)$ on variance. Numerical studies support these findings and suggest practical insights.