Despite the growing success of transform-based tensor models such as the t-product, their underlying geometric principles remain poorly understood. Classical differential geometry, built on real-valued function spaces, is not well suited to capture the algebraic and spectral structure induced by transform-based tensor operations. In this work, we take an initial step toward a geometric framework for tensors equipped with tube-wise multiplication via orthogonal transforms. We introduce the notion of smooth t-manifolds, defined as topological spaces locally modeled on structured tensor modules over a commutative t-scalar ring. This formulation enables transform-consistent definitions of geometric objects, including metrics, gradients, Laplacians, and geodesics, thereby bridging discrete and continuous tensor settings within a unified algebraic-geometric perspective. On this basis, we develop a statistical procedure for testing whether tensor data lie near a lowdimensional t-manifold, and provide nonasymptotic guarantees for manifold fitting under noise. We further establish approximation bounds for tensor neural networks that learn smooth functions over t-manifolds, with generalization rates determined by intrinsic geometric complexity. This framework offers a theoretical foundation for geometry-aware learning in structured tensor spaces and supports the development of models that align with transform-based tensor representations.

Monocular 3D tracking aims to capture the long-term motion of pixels in 3D space from a single monocular video and has witnessed rapid progress in recent years. However, we argue that the existing monocular 3D tracking methods still fall short in separating the camera motion from foreground dynamic motion and cannot densely track newly emerging dynamic subjects in the videos. To address these two limitations, we propose TrackingWorld, a novel pipeline for dense 3D tracking of almost all pixels within a world-centric 3D coordinate system. First, we introduce a tracking upsampler that efficiently lifts the arbitrary sparse 2D tracks into dense 2D tracks. Then, to generalize the current tracking methods to newly emerging objects, we apply the upsampler to all frames and reduce the redundancy of 2D tracks by eliminating the tracks in overlapped regions. Finally, we present an efficient optimization-based framework to back-project dense 2D tracks into world-centric 3D trajectories by estimating the camera poses and the 3D coordinates of these 2D tracks. Extensive evaluations on both synthetic and real-world datasets demonstrate that our system achieves accurate and dense 3D tracking in a world-centric coordinate frame.

Recent advances in foundational video generators (33; 38; 28), particularly through large diffusion transformers, have significantly improved video generation capabilities. This progress naturally suggests leveraging these powerful foundation models to advance video inpainting and editing. However, effectively utilizing their conditional generation abilities for these tasks would typically demand substantial computational resources for training, given their massive scale. Furthermore, as foundation models continue to evolve, traditional approaches relying on extensive fine-tuning will face increasing challenges in adapting to new video generators. An alternative solution is to employ these video generators as data priors, enabling task resolution in a training-free manner. Recent research has extensively explored methods for enabling conditional generation in image diffusion models.

Continuous time systems are often modeled using discrete time dynamics but this requires a small simulation step to maintain accuracy. In turn, this requires a large planning horizon which leads to computationally demanding planning problems and reduced performance. Previous work in model-free reinforcement learning has partially addressed this issue using action repeats where a policy is learned to determine a discrete action duration. Instead we propose to control the continuous decision timescale directly by using temporally-extended actions and letting the planner treat the duration of the action as an additional optimization variable along with the standard action variables. This additional structure has multiple advantages. It speeds up simulation time of trajectories and, importantly, it allows for deep horizon search in terms of primitive actions while using a shallow search depth in the planner. In addition, in the model-based reinforcement learning (MBRL) setting, it reduces compounding errors from model learning and improves training time for models. We show that this idea is effective and that the range for action durations can be automatically selected using a multi-armed bandit formulation and integrated into the MBRL framework. An extensive experimental evaluation both in planning and in MBRL, shows that our approach yields faster planning, better solutions, and that it enables solutions to problems that are not solved in the standard formulation.

Diffusion models have achieved remarkable success across a wide range of generative tasks. A key challenge is understanding the mechanisms that prevent their memorization of training data and allow generalization. In this work, we investigate the role of the training dynamics in the transition from generalization to memorization. Through extensive experiments and theoretical analysis, we identify two distinct timescales: an early time τgen at which models begin to generate high-quality samples, and a later time τmem beyond which memorization emerges. Crucially, we find that τmem increases linearly with the training set size n, while τgen remains constant. This creates a growing window of training times with n where models generalize effectively, despite showing strong memorization if training continues beyond it. It is only when nbecomes larger than a model-dependent threshold that overfitting disappears at infinite training times. These findings reveal a form of implicit dynamical regularization in the training dynamics, which allow to avoid memorization even in highly overparameterized settings. Our results are supported by numerical experiments with standard U-Net architectures on realistic and synthetic datasets, and by a theoretical analysis using a tractable random features model studied in the high-dimensional limit.

The rapid emergence of diverse large language models (LLMs) has spurred the development of LLM routers that assign user queries to the most suitable model. However, existing LLM routers typically perform a single-round, one-to-one mapping (i.e., assigning each query to a single model in isolation), which limits their capability to tackle complex tasks that demand the complementary strengths of multiple LLMs. In this paper, we present Router-R1, a reinforcement learning (RL)-based framework that formulates multi-LLM routing and aggregation as a sequential decision process. Router-R1 instantiates the router itself as a capable LLM, leveraging its reasoning ability to interleave "think" actions (internal deliberation) with "route" actions (dynamic model invocation), and integrates each response into its evolving context. To facilitate learning, we employ a lightweight rule-based reward comprising format rewards, final outcome rewards, and a novel cost reward for optimizing the balance between performance and cost, opening a pathway toward enhancing performance-cost trade-offs via RL.

The core task of recommender systems is to learn user preferences from historical user-item interactions. With the rapid development of large language models (LLMs), recent research has explored leveraging the reasoning capabilities of LLMs to enhance rating prediction tasks. However, existing distillation-based methods suffer from limitations such as the teacher model's insufficient recommendation capability, costly and static supervision, and superficial transfer of reasoning ability. To address these issues, this paper proposes RecZero, a reinforcement learning (RL)-based recommendation paradigm that abandons the traditional multi-model and multi-stage distillation approach. Instead, RecZero trains a single LLM through pure RL to autonomously develop reasoning capabilities for rating prediction. RecZero consists of two key components: (1) "Think-before-Recommendation" prompt construction, which employs a structured reasoning template to guide the model in step-wise analysis of user interests, item features, and user-item compatibility; and (2) rule-based reward modeling, which adopts group relative policy optimization (GRPO) to compute rewards for reasoning trajectories and optimize the LLM. Additionally, the paper explores a hybrid paradigm, RecOne, which combines supervised fine-tuning with RL, initializing the model with coldstart reasoning samples and further optimizing it with RL. Experimental results demonstrate that RecZero and RecOne significantly outperform existing baseline methods on multiple benchmark datasets, validating the superiority of the RL paradigm in achieving autonomous reasoning-enhanced recommender systems.

High-quality supervised finetuning (SFT) data are essential for unlocking pretrained LLMs' capabilities. Typically, instructions are paired with responses from various sources by humans annotators or other LMs, which are often out of the distribution of the target model to be finetuned. This, at scale, can lead to diminishing returns and even hurt the models' performance and robustness. We hypothesize that SFT is most effective with data aligned to the model's pretrained distribution and propose GRAPE-- a novel SFT framework that tailors supervision to the target model. For each instruction, it gathers responses from various sources, and selects the one that aligns most closely to the target model's pretrained distribution, as measured by the normalized probability. We then proceed with standard SFT with these selected responses. We first evaluate GRAPE with a controlled experiment, where we sample various solutions for each question in UltraInteract from multiple models and finetune on GRAPE-selected data using LMs from different families including LLaMA.1-8B,

Strategic classification, where individuals modify their features to influence machine learning (ML) decisions, presents critical fairness challenges. While group fairness in this setting has been widely studied, individual fairness remains underexplored. We analyze threshold-based classifiers and prove that deterministic thresholds violate individual fairness. Then, we investigate the possibility of using a randomized classifier to achieve individual fairness. We introduce conditions under which a randomized classifier ensures individual fairness and leverage these conditions to find an optimal and individually fair randomized classifier through a linear programming problem. Additionally, we demonstrate that our approach can be extended to group fairness notions. Experiments on real-world datasets confirm that our method effectively mitigates unfairness and improves the fairness-accuracy trade-off.

In recent advancements in Test Time Adaptation (TTA), most existing methodologies focus on updating normalization layers to adapt to the test domain. However, the reliance on normalization-based adaptation presents key challenges. First, normalization layers such as Batch Normalization (BN) are highly sensitive to small batch sizes, leading to unstable and inaccurate statistics. Moreover, normalizationbased adaptation is inherently constrained by the structure of the pre-trained model, as it relies on training-time statistics that may not generalize well to unseen domains. These issues limit the effectiveness of normalization-based TTA approaches, especially under significant domain shift.