Inverse problems are fundamental to science and engineering, where the goal is to infer an underlying signal or state from incomplete or noisy measurements. Recent approaches employ diffusion models as powerful implicit priors for such problems, owing to their ability to capture complex data distributions. However, existing diffusion-based methods for inverse problems often rely on strong approximations of the posterior distribution, require computationally expensive gradient backpropagation through the score network, or are restricted to linear measurement models. In this work, we propose Restart for Posterior Sampling (RePS), a general and efficient framework for solving both linear and non-linear inverse problems using pre-trained diffusion models. RePS builds on the idea of restart-based sampling, previously shown to improve sample quality in unconditional diffusion, and extends it to posterior inference. Our method employs a conditioned ODE applicable to any differentiable measurement model and introduces a simplified restart strategy that contracts accumulated approximation errors during sampling. Unlike some of the prior approaches, RePS avoids backpropagation through the score network, substantially reducing computational cost. W e demonstrate that RePS achieves faster convergence and superior reconstruction quality compared to existing diffusion-based baselines across a range of inverse problems, including both linear and non-linear settings.

Gaining sustainable performance improvement with scaling data and model budget remains a pivotal yet unresolved challenge in autonomous driving. While autore-gressive models exhibited promising data-scaling efficiency in planning tasks, predicting ego trajectories alone suffers sparse supervision and weakly constrains how scene evolution should shape ego motion. Therefore, we introduce DAP, a discrete-token autoregressive planner that jointly forecasts BEV semantics and ego trajectories, thereby enforcing comprehensive representation learning and allowing predicted dynamics to directly condition ego motion. In addition, we incorporate a reinforcement-learning-based fine-tuning, which preserves supervised behavior cloning priors while injecting reward-guided improvements. Despite a compact 160M parameter budget, DAP achieves state-of-the-art performance on open-loop metrics and delivers competitive closed-loop results on the NA VSIM benchmark. Overall, the fully discrete-token autoregressive formulation operating on both rasterized BEV and ego actions provides a compact yet scalable planning paradigm for autonomous driving.

In this paper, we explore the two-point zeroth-order gradient estimator and identify the distribution of random perturbations that minimizes the estimator's asymptotic variance as the perturbation stepsize tends to zero. We formulate it as a constrained functional optimization problem over the space of perturbation distributions. Our findings reveal that such desired perturbations can align directionally with the true gradient, instead of maintaining a fixed length. While existing research has largely focused on fixed-length perturbations, the potential advantages of directional alignment have been overlooked. To address this gap, we delve into the theoretical and empirical properties of the directionally aligned perturbation (DAP) scheme, which adaptively offers higher accuracy along critical directions. Additionally, we provide a convergence analysis for stochastic gradient descent using δ -unbiased random perturbations, extending existing complexity bounds to a wider range of perturbations. Through empirical evaluations on both synthetic problems and practical tasks, we demonstrate that DAPs outperform traditional methods under specific conditions. Zeroth-order optimization (ZOO) has emerged as a crucial paradigm in machine learning and optimization, particularly in scenarios where gradient information is unavailable or prohibitively expensive to compute. The randomized method (Akhavan et al., 2022) has also emerged as a critical direction. While traditional first-order methods utilize the stochastic gradient f p x; ξ q to update parameters, zeroth-order optimization relies solely on function evaluations.

Dataset distillation aims to synthesize compact yet informative datasets from large ones. A significant challenge in this field is achieving a trifecta of diversity, generalization, and representativeness in a single distilled dataset. Although recent generative dataset distillation methods adopt powerful diffusion models as their foundation models, the inherent representativeness prior in diffusion models is overlooked. Consequently, these approaches often necessitate the integration of external constraints to enhance data quality. To address this, we propose Diffusion As Priors (DAP), which formalizes representativeness by quantifying the similarity between synthetic and real data in feature space using a Mercer kernel. We then introduce this prior as guidance to steer the reverse diffusion process, enhancing the representativeness of distilled samples without any retraining. Extensive experiments on large-scale datasets, such as ImageNet-1K and its subsets, demonstrate that DAP outperforms state-of-the-art methods in generating high-fidelity datasets while achieving superior cross-architecture generalization. Our work not only establishes a theoretical connection between diffusion priors and the objectives of dataset distillation but also provides a practical, training-free framework for improving the quality of the distilled dataset.

Enhancing the mathematical reasoning capabilities of LLMs has garnered significant attention in both the mathematical and computer science communities. Recent works have made substantial progress in both Natural Language (NL) reasoning and Formal Language (FL) reasoning by leveraging the potential of pure Reinforcement Learning (RL) methods on base models. However, RL approaches struggle to impart new capabilities not presented in the base model, highlighting the need to integrate more knowledge like FL into NL math reasoning effectively. Yet, this integration is challenging due to inherent disparities in problem structure and reasoning format between NL and FL. To address these challenges, we introduce **NL-FL HybridReasoning (NFL-HR)**, an end-to-end framework designed to incorporate the FL expert into NL math problem-solving. To bridge the NL and FL input format gap, we propose the NL-FL Problem Alignment method, which reformulates the Question-Answering (QA) problems in NL as existence theorems in FL. Subsequently, the Mixed Problem Input technique we provide enables the FL reasoner to handle both QA and existence problems concurrently. Lastly, we mitigate the NL and FL output format gap in reasoning through an LLM-based Answer Extraction mechanism. Comprehensive experiments demonstrate that the NFL-HR framework achieves **89.80**% and **84.34%** accuracy rates on the MATH-500 and the AMC benchmarks, surpassing the NL baseline by **4.60%** and **4.82%**, respectively. Notably, some problems resolved by our framework remain unsolved by the NL baseline model even under a larger number of trials.

Real-world networks grow over time; statistical models based on node exchangeability are not appropriate. Instead of constraining the structure of the \textit{distribution} of edges, we propose that the relevant symmetries refer to the \textit{causal structure} between them. We first enumerate the 96 causal directed acyclic graph (DAG) models over pairs of nodes (dyad variables) in a growing network with finite ancestral sets that are invariant to node deletion. We then partition them into 21 classes with ancestral sets that are closed under node marginalization. Several of these classes are remarkably amenable to distributed and asynchronous evaluation. As an example, we highlight a simple model that exhibits flexible power-law degree distributions and emergent phase transitions in sparsity, which we characterize analytically. With few parameters and much conditional independence, our proposed framework provides natural baseline models for causal inference in relational data.

The development of fair and unbiased machine learning models remains an ongoing objective for researchers in the field of artificial intelligence. We introduce the Differential Adjusted Parity (DAP) loss to produce unbiased informative representations. It utilises a differentiable variant of the adjusted parity metric to create a unified objective function. By combining downstream task classification accuracy and its inconsistency across sensitive feature domains, it provides a single tool to increase performance and mitigate bias. A key element in this approach is the use of soft balanced accuracies. In contrast to previous non-adversarial approaches, DAP does not suffer a degeneracy where the metric is satisfied by performing equally poorly across all sensitive domains. It outperforms several adversarial models on downstream task accuracy and fairness in our analysis. Specifically, it improves the demographic parity, equalized odds and sensitive feature accuracy by as much as 22.5\%, 44.1\% and 40.1\%, respectively, when compared to the best performing adversarial approaches on these metrics. Overall, the DAP loss and its associated metric can play a significant role in creating more fair machine learning models.

This paper presents Dual Action Policy (DAP), a novel approach to address the dynamics mismatch inherent in the sim-to-real gap of reinforcement learning. DAP uses a single policy to predict two sets of actions: one for maximizing task rewards in simulation and another specifically for domain adaptation via reward adjustments. This decoupling makes it easier to maximize the overall reward in the source domain during training. Additionally, DAP incorporates uncertainty-based exploration during training to enhance agent robustness. Experimental results demonstrate DAP's effectiveness in bridging the sim-to-real gap, outperforming baselines on challenging tasks in simulation, and further improvement is achieved by incorporating uncertainty estimation.

Solving storage problem: where objects must be accurately placed into containers with precise orientations and positions, presents a distinct challenge that extends beyond traditional rearrangement tasks. These challenges are primarily due to the need for fine-grained 6D manipulation and the inherent multi-modality of solution spaces, where multiple viable goal configurations exist for the same storage container. We present a novel Diffusion-based Affordance Prediction (DAP) pipeline for the multi-modal object storage problem. DAP leverages a two-step approach, initially identifying a placeable region on the container and then precisely computing the relative pose between the object and that region. Existing methods either struggle with multi-modality issues or computation-intensive training. Our experiments demonstrate DAP's superior performance and training efficiency over the current state-of-the-art RPDiff, achieving remarkable results on the RPDiff benchmark. Additionally, our experiments showcase DAP's data efficiency in real-world applications, an advancement over existing simulation-driven approaches. Our contribution fills a gap in robotic manipulation research by offering a solution that is both computationally efficient and capable of handling real-world variability. Code and supplementary material can be found at: https://github.com/changhaonan/DPS.git.

Many spatiotemporal domains handle multi-agent trajectory data, but in real-world scenarios, collected trajectory data are often partially missing due to various reasons. While existing approaches demonstrate good performance in trajectory imputation, they face challenges in capturing the complex dynamics and interactions between agents due to a lack of physical constraints that govern realistic trajectories, leading to suboptimal results. To address this issue, the paper proposes a Derivative-Based Hybrid Prediction (DBHP) framework that can effectively impute multiple agents' missing trajectories. First, a neural network equipped with Set Transformers produces a naive prediction of missing trajectories while satisfying the permutation-equivariance in terms of the order of input agents. Then, the framework makes alternative predictions leveraging velocity and acceleration information and combines all the predictions with properly determined weights to provide final imputed trajectories. In this way, our proposed framework not only accurately predicts position, velocity, and acceleration values but also enforces the physical relationship between them, eventually improving both the accuracy and naturalness of the predicted trajectories. Accordingly, the experiment results about imputing player trajectories in team sports show that our framework significantly outperforms existing imputation baselines.