生成模型理论与方法

Multi-modal spatio-temporal forecasting (MM-STF) supports weather nowcasting, traffic prediction, and earth-system modeling by combining heterogeneous sources such as physical fields, satellite imagery, and in-situ sensors. Three obstacles persist: (i) modalities have different spatio-temporal sampling rates, forcing lossy interpolation onto a unified grid; (ii) modalities are frequently missing at deployment due to sensor outages or revisit gaps, while most methods train with full availability; and (iii) autoregressive decoders accumulate errors over long horizons, amplified by multi-modal conditioning. We propose AsyncCouple-Flow to address these issues jointly. A Modality-Aware Token Sparsification (MATS) module performs scale-aware tokenization and uses a shared importance scorer to select top-k tokens per timestep, producing equal-length sequences. An Asynchronous Cross-Modal Coupling Graph (ACCG) replaces fixed cross-attention with a learnable graph whose edges encode time offsets, semantic similarity, and modality-specific physical priors, enabling fusion under arbitrary asynchrony and missingness. A Flow-Matching Forecasting Head models multi-step prediction as a conditional ODE, trained with stochastic modality dropout and integrated jointly to avoid autoregressive drift. Experiments on ERA5+GOES+ISD weather forecasting and PEMS-BAY traffic prediction with multi-source side information show that AsyncCouple-Flow outperforms state-of-the-art baselines and remains robust with up to two missing modalities. The code will be released upon acceptance.

In flow matching (FM), a velocity model $v_θ$ is trained using a predefined path $g_t$ that connects data and noise samples (e.g., $g_t(x_0, x_1) = (1 - t) x_0 + t x_1$). In this work, we study the choice of this path from an optimization perspective by analyzing the variance of stochastic gradients. We consider the class $G(p_t,v^\star_t)$ of paths that induce the same marginal distributions $p_t$ and marginal velocity field $v^\star_t$, and therefore the same FM objective. Our main finding is that the choice of path $g_t$ can fundamentally change the convergence rate of SGD, even when the FM objective remains exactly the same. (i) For a linear velocity model and one-dimensional Gaussian data, we derive a tight bound on the SGD iteration complexity up to logarithmic factors and find an analytically optimal path that minimizes this bound among linear paths inducing the same FM problem. (ii) We then extend the variance analysis to general FM problems and formulate path selection at a fixed $θ$ as the variance-minimization problem PathOpt$_θ$, constrained to $g_t\in G(p_t,v^\star_t)$. We show that this constraint is essential: reducing variance without it can lead to slower convergence. (iii) Since the constraint $g_t \in G(p_t,v^\star_t)$ cannot generally be verified directly, we derive an equivalent formulation with constraints that can be estimated from samples, allowing paths to be found numerically. Our theoretical results are supported by experiments with Gaussian data, Gaussian mixture models, and real datasets.

Reducing radiation exposure in Positron Emission Tomography (PET) is important for patient safety; however, ultra-low-dose imaging suffers from severe noise, which may affect diagnostic interpretation without appropriate image enhancement. While current 3D deep generative models, particularly diffusion models, have shown strong reconstruction fidelity, their practical use can be limited by long inference times. In contrast, faster 2D-based alternatives may have difficulty maintaining volumetric consistency, an important consideration for whole-body PET imaging analysis. To bridge this gap, we propose a one-pass conditional 3D rectified flow (3D Flow) framework for whole-body PET image denoising that incorporates a novel optimized non-uniform sampling strategy. The model is trained with a one-pass linear-interpolant velocity-matching objective. This approach reconstructs a full 3D volume in approximately 30 seconds in our implementation, compared with multi-hour inference for the evaluated 3D DDPM baseline. Evaluations including zero-shot transfer to an independent clinical dataset show that our model achieves favorable global image quality and lesion conspicuity compared with the evaluated 3D DDPM and DDIM baselines, including on challenging short-acquisition data. Furthermore, the proposed method shows promising zero-shot transfer performance across the evaluated datasets and unseen dose levels (down to 1/100 of the standard dose), with artifact-focused visual comparisons supporting the need for further lesion-level validation. By balancing reconstruction fidelity and computational efficiency, this work presents a candidate approach for ultra-low-dose whole-body PET image denoising.

Diffusion large language models (dLLMs) offer a promising parallel decoding paradigm as an alternative to autoregressive generation through iterative unmasking. However, dLLMs typically require many steps before token confidence reaches the decoding threshold, resulting in inefficient inference even with block-wise KV caching. To accelerate dLLM inference, we for the first time propose an "early-bird (EB)" decoding framework, motivated by the observation that tokens with similarly low entropy tend to cluster and can be jointly decoded earlier, before reaching the confidence threshold. In particular, our EB-Decode framework integrates two key enablers: (1) a learnable network that adaptively groups tokens with similar uncertainty into variable-length blocks, rather than relying on fixed block sizes; (2) a position-aware sampler that learns to unmask tokens in parallel using fewer decoding steps within predicted variable-length blocks. Both components are developed without modifying pretrained dLLM weights and can therefore be directly deployed as plug-ins during serving, with negligible training and inference overhead. Extensive experiments across three models and four benchmarks consistently validate our observation and the effectiveness of EB-Decode, achieving 3.53-18.76$\times$ higher throughput than the vanilla decoding method and up to 1.58$\times$ higher throughput over the strongest baseline, Fast-dLLM, with comparable accuracy.

State Space Models (SSMs) have emerged as a compelling alternative to Transformers, enabling sequence modeling with constant memory and linear compute. Although SSMs exhibit reasonable performance and favorable computational characteristics, they continue to lag behind Transformers on tasks that require in-context learning and precise retrieval, slowing their adoption for large-scale language modeling. In this work, we demonstrate that both the success and failure of SSMs in these domains can be explained by studying the role of the gating mechanism, a prevalent component in modern recurrent networks. Specifically, we show through theory and experiments that this gating mechanism causes SSMs to first learn an in-weights "memorization" solution, while delaying, or even preventing, convergence to a correct in-context learning solution. Importantly, this happens even in cases where there are no fundamental limitations due to the architecture or its memory capacity. On the other hand, we find that gating is often beneficial for improving generalization to long sequence lengths. Our results illuminate the crucial role of the gating mechanism in shaping both the training dynamics and generalization of SSMs, and provide a basis for understanding and improving linear-time models.

Vision-language models (VLMs) and vision-language-action models (VLAs) are increasingly deployed in real-world applications. There, a small perturbation to the recorded camera image may change a decision significantly. However, existing benchmarks for these models only sample perturbations, which does not guarantee the absence of a failure in the untested region. We present the first robustness validation of six VLMs (drawn from the Gemma, InternVL, LLaVA, and Qwen families) and five VLAs (drawn from the GR00T, OpenVLA, and $π$ families) over entire continuous regions of photometric and geometric image perturbation: brightness shifts, camera rotations, and their composition. To this end, we build on the validation framework H$^2$V and introduce H$^2$V-M, a margin-aware convergence rule that makes validation affordable at the 32B parameter scale. We demonstrate that H$^2$V-M outperforms H$^2$V by an order of magnitude in model queries and that it finds counterexamples faster than random sampling while providing soundness guarantees. Our VLM and VLA robustness validation shows that robustness is mostly dependent on the perturbation type, rather than the model, and that VLMs are more robust to large camera rotations than VLAs. For VLAs, even perturbations as small as $\pm1^\circ$ can change the commanded action in many cases. We also show that robustness depends more on model family than on model size.

Goldwasser et al. showed that undetectable backdoors can be planted in machine learning models trained with the Random Fourier Features (RFF) algorithm, under a hardness assumption tied to the Continuous Learning With Errors (CLWE) problem. Under standard cryptographic assumptions, even a full white-box audit of a model's weights cannot detect this class of backdoor. The construction is stated in terms of cryptographic reductions and probabilistic lemmas, without a reference implementation, and relies on secondary machinery such as the Sparse Gaussian Pancakes distribution and a homogeneous CLWE conditional density. Its realizability in ordinary numerical code is not obvious from the paper alone. This paper implements the white-box CLWE-RFF backdoor construction end to end using only numpy and scipy, to test whether this threat is realizable with commodity scientific-computing tools or requires specialized cryptographic infrastructure. We give two samplers for the core $GP_d(b_k)$ distribution. The first is a rejection-sampling proxy. The second is an exact closed-form sampler derived from the homogeneous CLWE density and verified against its own analytic form. Using this implementation, we run statistical indistinguishability tests, covering both weight-space and functional black-box comparisons. We find no evidence of detectable difference between backdoored and clean models across a range of sparsity ratios $ρ= d_{\text{sparse}}/D$. We report which parts of the construction were straightforward to realize, which required derivation not spelled out in the paper. We also highlight which parts we did not attempt to reproduce, including the underlying lattice hardness reduction. We see this work as a contribution to understanding the practical realizability of the Goldwasser white-box CLWE core, not as a new theoretical result.

Robust aggregation methods for federated learning quietly rest on a fragile assumption: that whoever shows up in a given round is a fair sample of the full population. In practice, they rarely are. When only a handful of clients participate per round, even a modest fraction of adversaries can dominate that sample and silently invalidate the finite-sample guarantees that coordinate-wise median, Krum, Bulyan, and trimmed mean all depend on. We introduce SWB-DM to address this directly. SWB treats each slice of a client update as a one-dimensional distribution, computes a trimmed Wasserstein barycenter across clients, and recovers coordinate identity via a medoid-based gauge-fixing step — a heuristic we developed and do not claim it belongs to standard optimal-transport theory. DeMoA-style delayed momentum then caches updates across the full client population each round, decoupling robustness from whoever happened to be sampled. Trim ratio calibration is not cosmetic: under-trimming causes collapse at corruption levels a properly calibrated model survives. Across 448 CIFAR-10 configurations, plus CIFAR-100, FEMNIST, and a 500-client scalability run, we find several mechanistically distinct failure modes. Even-sample coordinate-wise median degrades to a deterministic wrong answer. Krum silently violates its own n greater than 2f+2 precondition and diverges without warning. Bulyan's n greater than or equal to 4f+3 threshold produces a sharp pass/fail boundary. On attacks, IPM defeats order-statistic defenses — including SWB — more reliably than ALIE, confirmed through delta-space measurements against a convergence bound. SWB-DM's cache carries a real warm-up cost, but extending all baselines to the same round budget shows its CIFAR-10 gains are disproportionately large. On CIFAR-100, FLTrust benefits more — for reasons entirely unrelated to caching.

This letter presents a unified formulation and a controlled numerical comparison of generative-model approaches to the nonlinear filtering problem. Under this formulation the analysis step is realized by a transport of the forecast distribution to the posterior, the approaches differing only in how that transport is selected and learned. We derive three new filters, based on stochastic interpolants, their deterministic flow-matching limit, and Schrödinger bridges realized through forward--backward SDEs. We develop a two-stage tuning procedure that separates the training of the generative model from its online refinement. The resulting methods are compared against the optimal transport filter (OTF), the Knothe--Rosenblatt filter (KRF), the sequential importance resampling (SIR) particle filter and the ensemble Kalman filter (EnKF), in terms of accuracy, computational time, and sensitivity to ensemble size and state dimension. The results indicate that every generative filter resolves multimodal posteriors that the EnKF and SIR do not, that no single generative framework dominates, the preferred method being set by the available online budget and ensemble size, and that the filters differ in the regularity of the particle trajectories they produce.

Amid the rapid advancement of physical-world intelligence, cloud-edge collaborative large language models (LLMs) have emerged as a promising roadmap for practical LLM deployment. However, existing cloud-edge paradigms struggle to balance global consensus with local personalization, which fails to satisfy the need for a unified knowledge foundation on the cloud and domain-specific adaptation at the edge. To address this, we introduce, for the first time, a personalized bilevel optimization framework that formalizes cloud-edge LLM collaboration as a dual structure: the upper level optimizes edge-side personalization, while the lower level governs cloud-side knowledge transfer, reaching cloud-edge evolving in coordination. We then propose CIDERS, an efficient solver that decomposes the model into a learnable backbone and a messenger. While the cloud performs knowledge transfer to the learnable backbone, the key lies in embedding global trajectories into each local personalization step via consensus-variate correction to reconcile personalization with consensus. We provide a comprehensive theoretical analysis, including a geometric characterization of the local trajectory and a full convergence guarantee, revealing an explicit trade-off structure between personalization and global convergence. Extensive experiments demonstrate that CIDERS consistently outperforms competitive baselines on the compressed edge path, with 3.1x and 1.7x gains on mathematical reasoning and code generation, respectively, and a 10% relative gain on instruction metrics. Mechanism experiments attribute these gains to early consensus-corrected coordination and task-aware distillation. Overall, CIDERS offers a viable path toward consensus-guided continuous personalization in cloud-edge LLM systems.

Traditional variance reduction methods (e.g., SPIDER, SARAH, STORM) have been extensively investigated for improving the convergence rates of stochastic optimization. These techniques typically maintain a sequence of estimators for a single function (or gradient) across iterations. However, what if we need to track multiple functions, but can only access stochastic samples of $\mathcal{O}(1)$ functions at each iteration? This scenario arises in an important emerging family of finite-sum coupled compositional optimization (FCCO) problems of the form $\frac{1}{m}\sum_{i=1}^m f_i(g_i(\mathbf{w}))$, where each $g_i$ is accessible only through a stochastic oracle. The key challenge is to track $\mathbf g(\mathbf{w})=(g_1(\mathbf{w}), \ldots, g_m(\mathbf{w}))$ over time, where $\mathbf g(\mathbf{w})$ has $m$ blocks but only $\mathcal{O}(1)$ blocks can be probed for their stochastic values at each step. To address this challenge, we propose a novel Multi-block-Single-probe Variance Reduction (MSVR) estimator to efficiently trace $\mathbf g(\mathbf{w})$ under partial block sampling. Building on the MSVR estimator, we develop several algorithms for FCCO problems, achieving improved sample complexities for non-convex, convex, strongly convex, and Polyak-Łojasiewicz (PL) objectives. We further obtain an improved dependence on $m$ when the outer function gradients $\nabla f_i$ are linear. Empirical studies on multi-task deep AUC maximization further demonstrate the superior performance of the proposed estimators.

For constant-stepsize stochastic approximation (SA), the iterates converge in distribution to a stationary law that depends on the stepsize $α.$ Steady-state convergence (SSC) concerns the limit of the scaled stationary distribution as $α\downarrow 0.$ Existing SSC theory requires i.i.d. or additive noise and global differentiability of the mean operator, and yields suboptimal rates. We develop a unified SSC theory for constant-stepsize contractive SA driven by Markovian, multiplicative noise, covering both locally differentiable and locally nondifferentiable mean operators. A key methodological contribution is a multi-step universality framework that progressively reduces the original stochastic recursion to tractable auxiliary dynamics while preserving its steady-state limit. Under local quadratic linearization at the fixed point, we obtain a Gaussian approximation of the scaled steady state at the optimal rate $O(\sqrtα)$ in Wasserstein-2 distance, which further gives finite-time Gaussian approximations for the raw iterates. In the locally nondifferentiable regime, we establish a general SSC result and show that the leading-order asymptotic bias can be of order $\sqrtα$, in contrast to the $α$-order bias in the smooth regime. We apply the theory to Markovian linear SA and asynchronous Q-learning, neither of which is covered by prior results. We further propose a bias-reduction scheme for Q-learning that requires no knowledge of the local smoothness regime, validated by numerical experiments.

We study density ratio estimation and importance-weighted regression under target shift with continuous outputs. Under target shift, the conditional distribution of the inputs given the outputs remains invariant across the training and test distributions, while the output marginal distribution may change. Although this problem has been extensively studied for discrete outputs, the continuous setting is substantially less understood: the importance weights are determined by an unknown density ratio function, for which existing estimation methods lack explicit finite-sample convergence rates. We propose a spectral regularization method in a reproducing kernel Hilbert space (RKHS) for estimating the continuous density ratio from labeled training samples and unlabeled test inputs. Under a source condition with regularity parameter $ι>0$, we establish high-probability finite-sample guarantees and show that the estimator achieves the capacity-independent minimax-optimal RKHS-norm rate $O(n_η^{-ι/(2ι+2)})$. We then incorporate the estimated density ratio into importance-weighted regression and characterize the propagation of density-ratio estimation error to the final predictor. When sufficiently many samples are available for density ratio estimation, the resulting regression estimator attains the minimax-optimal rates of standard kernel regression. These results establish a finite-sample theory for continuous density ratio estimation and importance-weighted learning under target shift.

Missing values are ubiquitous in heterogeneous data mining, where numerical, categorical, and binary variables often coexist. Many imputation methods, especially diffusion-based ones, treat discrete variables through continuous surrogates such as one-hot relaxations rather than modeling them natively. This creates a mismatch between the model state space and the mixed discrete and continuous structure of the data. We propose Impute-EM, an Expectation Maximization style framework that alternates between imputing missing entries with the current model and refitting a diffusion backbone on completed data. We instantiate Impute-EM with native mixed-state diffusion backbones for heterogeneous data, combining Gaussian and masked categorical components without one-hot relaxations. In exact settings, we characterize the update and show that the observed mask-indexed marginals match the targets at the limit, while making explicit that the full data distribution is generally non-identifiable from incomplete observations alone. Empirically, Impute-EM delivers the best distributional fidelity on mixed-type tabular imputation, on which downstream modeling relies, with text imputation serving as a controlled validation of the native discrete backbone.

Diffusion models generate images through an iterative diffusion process, and recent studies have demonstrated that the intermediate feature maps produced during this process contain rich visual representations, leading to their adoption across a variety of downstream tasks. However, most existing approaches are limited to either using a single feature map at a specific timestep or aggregating feature maps across multiple timesteps. We observe that intermediate representations in the diffusion process form meaningful trajectories along the time axis. In particular, the representation of each spatial patch evolves progressively throughout the generative process, encoding semantics that are difficult to capture from static snapshots alone. This observation motivates the need to treat diffusion representations as temporally structured trajectories rather than static snapshots. To this end, we propose Diffusion Trajectory Modeling (DTM), a framework that interprets the temporal evolution of each spatial patch as a trajectory and leverages it for semantic correspondence. By effectively modeling patch-wise trajectories generated across multiple timesteps, DTM captures correspondence cues that prior methods are not designed to capture. We further demonstrate empirically that spatially corresponding patches form similar trajectory patterns throughout the diffusion process, suggesting that the temporal axis of diffusion carries semantic information. Experiments on SPair-71k, SPair-U and AP-10K show that DTM achieves strong performance, presenting a new perspective for exploiting diffusion representations from a trajectory-centric viewpoint.

Semantic watermarks for diffusion models embed ownership information into the generative process while preserving perceptual quality, but Gaussian-Shading extraction conventionally requires multi-step diffusion inversion to recover the initial noise. Recent one-step methods show that this cost can be reduced substantially. We study this problem through extended flow matching and conditional regression. The key observation is that, near the high-SNR image endpoint, recovering a useful noise statistic given by the first-step output of the extended flow matching in the high-SNR regime is much simpler than reconstructing the full inverse trajectory, and Gaussian Shading only requires the recovered latent to remain in the correct watermark decision region. Based on this observation, we propose a lightweight, prompt-free extractor that decomposes endpoint recovery into an image-like anchor and a noise-oriented residual, which increases the capability of the model to utilize GPU parallel computation. The resulting method avoids iterative inversion and repeated evaluation of a diffusion-scale U-Net, providing an efficient one-step extraction pipeline with a concise theoretical interpretation. The computational cost of extracting noise is lower than that of both OSI and FARI. The github repo is there: https://github.com/TheLovesOfLadyPurple/RAIN-lightweight-NN-for-one-step-semantic-watermark-extraction

Methods of Branched Optimal Transport (BOT) mimic the economy and efficiency of natural tree-like structures, such as those found in rivers and biological systems. These methods are widely applicable for designing efficient networks in society, from river basins and blood vessels to mail and gas distribution systems. However, they remain understudied in the context of designing deep generative models, particularly at a large scale. Standard continuous-time generative models, such as the flow matching approach, fail to capture the inherent hierarchical and branching patterns present in real-world data. Current models provide no mechanism for flows to merge or share pathways to minimize total transport cost. Inspired by the "economy of scale" principle in BOT, we introduce a novel, scalable branched flow-matching algorithm designed to solve the branched optimal transport problem in high dimensions. Our method adapts the Benamou-Brenier continuous-time optimal transport formulation to learn branched generative flows. These flows allow probability mass to aggregate along common pathways before branching out to diverse targets. Parametrized by neural networks, our method effectively learns complex branched generative processes. We demonstrate its effectiveness on challenging high-dimensional tasks in biology and image generation.

Drifting models offer a promising route to faster generative AI: they perform gradual transport during training, while generating new samples in a single step. This paper asks whether the underlying drifting process can converge rapidly to a target distribution under ideal conditions, before finite-data or optimization effects are introduced. We show that its convergence rate depends critically on how it handles spatial scale. With a single fixed resolution, fine-scale features of the target can become nearly invisible, leading to extremely slow convergence. We introduce a multihead approach that combines scale-normalized information across a continuum of resolutions. We prove that this multihead approach restores exponential convergence near standard reference distributions. These results identify fixed resolution as a key bottleneck and provide a simple route to faster one-step generative models.

Some of the sharpest challenges in sampling from the energy functions of physical systems arise at phase transitions, where the density of states changes abruptly and many sampling algorithms stall. Nested sampling is a particle method that traverses the density of states under a hard energy constraint and is known to be robust to such transitions, but its application in high dimension is limited by the difficulty of sampling under that constraint. In this work we introduce Quenched Ensemble Sampling, which generalises the hard constraint to a family of repulsive potentials at the energy boundary. This preserves the quenched path of monotonically decreasing energy while making the constrained target amenable to scalable gradient-based kernels. We demonstrate on synthetic models of phase transitions that our method estimates the marginal likelihood and draws posterior samples across a first-order transition where popular alternatives such as tempering fail. We apply the procedure to marginal likelihood estimation in Bayesian neural networks, enabling model comparison between network architectures. Finally, in a high-dimensional continuous lattice field theory, we show that this method traverses a first-order transition and estimates the partition function.

Flow-matching diffusion models have recently emerged as a strong paradigm for high-fidelity visual generation. However, their prohibitively high fine-tuning cost limits scalability to downstream tasks. While Low-Rank Adaptation (LoRA) combined with spectral initialization has demonstrated accelerated convergence and improved performance in autoregressive language models by better aligning gradient directions, we find that it fails to deliver similar gains in diffusion fine-tuning, often yielding marginal or even negative improvements over vanilla LoRA.We attribute this discrepancy to a fundamental mismatch between LoRA's low-rank parameterization and the intrinsically high-rank gradients induced by the flow-matching objective. In particular, stochastic timestep sampling introduces directionally heterogeneous gradient signals across training steps, leading to misaligned updates under low-rank constraints.To address this issue, we propose Prism-LoRA,a Principal-timestep Restricted Init via Sparse Matrix-decomposition framework that improves gradient alignment during fine-tuning. Our method consists of two key components: (i) principal timestep selection, which restricts initialization gradients to a subset of dominant timesteps to suppress effective gradient rank, and (ii) principal channel filtering, which removes task-irrelevant channels, enabling the one-step spectral initialization gradient to better align with the long-horizon optimization trajectory. Extensive experiments demonstrate that our method consistently improves both convergence speed and final performance across multiple diffusion fine-tuning benchmarks, including subject-driven generation, controllable generation, and deblurring, achieving not only performance improvement but also earlier stages of convergence over baseline LoRA and other spectral-init methods.

Pretrained score-based diffusion models provide strong unconditional priors, yet enforcing measurement or physics consistency in inverse problems is often handled by heuristic guidance, intermittent projections, or task-specific conditional training, with limited guarantees of feasibility at the end of inference. We propose terminal-conditioned inversion for score-based SDE priors. Given a frozen Score-SDE prior and a task-defined terminal feasibility specification, we construct an associated backward stochastic differential equation whose adapted solution defines a principled inverse map from the terminal requirement to a prior state at a chosen noise level. Under standard regularity conditions, we establish existence and uniqueness of the adapted solution and obtain terminal consistency by construction. We further develop a practical neural BSDE solver that composes arbitrary pretrained diffusion priors with domain constraints without modifying the score-defined coefficients, producing an anchored prior state that enables neighborhood sampling for uncertainty characterization. Experiments on toy datasets validate stable terminal-conditioned inversion and distributionally consistent neighborhood sampling. As a real-world case study, we apply the framework to sparse-view CT reconstruction and achieve improved reconstruction quality over representative training-free baselines while satisfying strict measurement feasibility under the prescribed terminal specification. Project is available in: \href{https://laplacelab.github.io/BSDEDiffusion/}{https://laplace.center/icmlbsdeI/}

Discrete diffusion and flow models are a promising alternative to autoregressive language models, but compressing many-step sampling into fewer steps typically requires distilling a pretrained teacher model. This caps the student at the teacher's quality and requires a costly two-stage training pipeline. We introduce Discrete Beckmann Transport Models (DBTM), built on a time-independent flow whose autonomous transport map provably carries any point in the ambient space to a fixed point on the vertices of the simplex in a single step. We show that this fixed-point property is characterized by a conservation equation whose residual can be minimized directly from data, removing the requirement for a teacher flow and time conditioning. Under this construction, a partially trained map corresponds to the flow truncated at finite time, so generation reduces to iterating one map until it reaches a fixed point. We further extend the map to a partial-context interpolant where additional function evaluations act as refinement steps rather than ODE integration steps. On language modeling and reasoning tasks, DBTM enables one- and few-step generation that improves quality and accuracy over discrete diffusion and continuous flow baselines.

Understanding the design space of streaming video diffusion is essential to exploring its potential for generation quality and computational efficiency. We develop a unified analytical framework that relates model and sampler choices, historical conditioning, execution scheduling, and training strategies. The framework accommodates a broad family of causal context-selection policies and makes their computational dependencies and training-inference alignment explicit. Within this design space, we study three representative policies: clean, same-level, and progressive history. On the full VBench prompt set, same-level and progressive history achieve aggregate scores of 85.24 and 85.60, respectively, compared with 84.45 for the clean-history reference. Long-video comparisons further show improved subject consistency and more coherent motion with progressive history. By allowing multiple denoising nodes to be processed together, progressive-history pipelining achieves $1.57$-$2.83\times$ steady-state DiT speedups under our evaluated conditions. We additionally find that LoRA adaptation of the DMD fake-score network improves generation quality using only 2.15% as many trainable fake-score parameters as full-parameter adaptation. Together, these findings show that fully denoised history is not a prerequisite for high-quality streaming generation and motivate the joint design of historical conditioning, execution, and training.

Constraining reaction rate coefficients is a central challenge in the development of explicit atmospheric chemical mechanisms, particularly for autoxidation systems where many reaction pathways are only indirectly observed through high-resolution mass spectrometry. In this study, we evaluate rate-coefficient optimisation methods for a toy-case autoxidation mechanism using synthetic data with known ground truth. Two complementary approaches are compared: ODE-constrained neural-network optimisation, which provides efficient point estimates of uncertain rate coefficients, and the Markov Chain Monte Carlo (MCMC) approach, which samples the posterior distribution of rate coefficients and quantifies parameter uncertainty. The methods are tested using direct concentration observations and mass-spectral observations under different noise levels. For unperturbed and low-noise synthetic observations, both methods converged towards the known rate coefficients, with the neural-network optimiser providing faster point estimates. Under high-noise conditions (with the signal-to-noise ratio approximately S / N = 1), however, MCMC was substantially more robust in recovering the rate coefficients. The posterior analysis shows that mass-spectral aggregation broadens credible intervals even at low noise, and that high-noise mass spectra can leave many individual reaction rates weakly identifiable. Posterior predictive validation nevertheless shows how broad parameter uncertainty constrained by MCMC remains consistent with accurate reproduction of the observable mass spectrum. These results demonstrate that point-estimation and Bayesian sampling methods provide complementary information: neural-network optimisation is effective for informative data, whereas MCMC is essential for diagnosing uncertainty, non-uniqueness, and identifiability in noisy or aggregated inverse problems.

Teaching abstract theoretical computer science (TCS) concepts such as algorithm analysis and complexity theory is challenging because students must handle formal proofs and asymptotic reasoning that conventional resources rarely explain in an adaptive, on-demand way. We present AlgoRAG, a specialized Retrieval-Augmented Generation (RAG) system that couples a large language model (LLM) with a curated, domain-specific knowledge base to address these challenges. The knowledge base integrates authoritative textbooks, 847 lecture slides, 312 practice problems with solutions, 156 worked proof templates, and 89 complexity worksheets. AlgoRAG incorporates domain-specific optimizations including mathematical entity recognition, notation-aware retrieval, and pedagogical re-ranking. We evaluate AlgoRAG on 179 curated exam-style questions spanning asymptotic analysis, recurrence relations, dynamic programming, graph algorithms, NP-completeness, sorting, and divide-and-conquer. The system achieves a 100% success rate with a mean response time of 38.0 seconds. While BLEU-4 scores are zero — a known limitation of n-gram matching on mathematical proofs where equivalent reasoning may use entirely different notation — AlgoRAG attains ROUGE-1 F1 of 0.0963, ROUGE-L F1 of 0.0683, and a pedagogical quality score of 0.7620, indicating that responses are well-structured and didactically sound even when surface wording diverges from reference answers. Performance is especially strong on NP-completeness (ROUGE-1 F1 = 0.1285, pedagogical quality = 0.7643) and graph algorithms (ROUGE-1 F1 = 0.1023, pedagogical quality = 0.8086). These results support the conclusion that RAG is an effective architecture for personalized theoretical-CS instruction, providing correct, context-rich explanations even for highly abstract topics.

Few-step distillation accelerates diffusion models but must balance diversity and fidelity: trajectory-based distillation preserves mode coverage, while distribution matching sharpens samples but can reduce diversity. We show that this tension can be exploited in a noise-regime-dependent way: high-noise steps largely determine global modes, whereas low-noise steps refine local details. We propose CrossDistill, a trajectory-level hybrid distillation framework that splits the sampling trajectory at a crossover point, applies a trajectory-preserving objective on the high-noise interval and a distribution-matching objective on the low-noise interval, and couples the two stages through the crossover state. In contrast to loss-level mixing, and complementarily to training-time two-stage recipes, CrossDistill explicitly assigns complementary objectives along the noise axis, so that global branching is preserved before local statistics are sharpened. CrossDistill is a noise-level scheduling policy: PCM and DMD are plug-in instantiations, while the noise partition, crossover coupling, and objective ordering are the key design elements. Experiments on text-to-video diffusion models and qualitative image-to-video results show that CrossDistill expands the few-step quality-diversity frontier, retaining seed-level variation while achieving competitive visual fidelity.

Reliable path tracking is crucial for autonomous underwater vehicles (AUVs) operating in dynamic and uncertain marine environments. However, traditional line-of-sight (LOS) guidance methods rely on asymptotic convergence, resulting in slow disturbance recovery and unpredictable tracking performance. Existing robust control methods typically require modifications to the underlying vehicle controller, limiting their practical application on commercial AUV platforms. This paper proposes a robust fixed-time adaptive LOS guidance framework for 3D path tracking for AUVs. By combining fixed-time stability theory with LOS guidance, this method guarantees path tracking convergence within a preset time range, with the convergence time independent of initial conditions. Furthermore, a fixed-time adaptive estimator is developed to rapidly compensate for time-varying sideslip disturbances caused by ocean currents. A time-varying look-ahead mechanism is also introduced to improve tracking performance on curved paths. Lyapunov analysis proves the fixed-time stability of the proposed framework, and numerical simulations and physical experiments demonstrate that, compared to state-of-the-art adaptive LOS methods, this framework exhibits superior tracking accuracy, convergence speed, and anti-interference capability. In simulation, the time-varying look-ahead variant reduced cross-track and vertical-track RMSE by 69.37% and 67.46%, respectively, during curved-path tracking. In field experiments with an Iver 3 AUV, the proposed fixed-time guidance reduced average tracking error by 56.35% in straight-path evaluation and 27.59% in curved-path evaluation compared with conventional adaptive LOS guidance.The proposed method provides a practical guidance-level solution for achieving reliable autonomous navigation of AUVs in complex marine environments.

Generation of musculoskeletal gait trajectories conditioned on patient-specific parameters remains a key challenge for wearable robotics and rehabilitation. Assistive systems such as lower-limb exoskeletons require reference trajectories that adapt to individual morphology and therapeutic goals while preserving biomechanical realism. Traditional approaches rely on hand-crafted gait templates or optimization procedures that scale poorly across subjects and walking conditions. In this work, we explore conditional diffusion models for generating lower-limb joint-angle trajectories conditioned on gait parameters such as step length. We compare a baseline transformer diffusion model with a controllable diffusion transformer variant incorporating adaptive normalization and classifier-free guidance. Experiments on a dataset of 4,590 gait cycles show that diffusion models can generate realistic periodic gait trajectories while enabling some controllability variation in gait characteristics, highlighting their potential for personalized gait synthesis in assistive robotics.

Training-free diffusion inverse solvers typically choose between local measurement guidance and costly clean-space posterior updates. Independent posterior refresh can improve global correction by sampling a clean conditional and re-noising it, but its practical realization requires probability-flow ODE integration and clean-space Markov chain Monte Carlo (MCMC). We propose Direct Conditional Transition Sampling (DCTS), a direct stochastic-flow approximation to the same ideal refresh target. Rather than explicitly drawing a clean sample, DCTS estimates the measurement-conditioned clean mean along a short inner path and transports Gaussian source noise directly to the next noisy state. A denoiser-compatible sufficient statistic and a covariance-scaled operator update enable this conditional-mean estimation. Experiments on four inverse problems demonstrate that DCTS achieves competitive reconstruction quality with up to $16.8\times$ speedups over competing methods.

Operator learning is formulated on function spaces, but training data are typically available only through finite-dimensional representations. In encoder--decoder architectures, a matrix-valued kernel on the encoded space induces an operator-valued kernel on the original function spaces, and the corresponding reproducing kernel Hilbert spaces are isometrically isomorphic. As the input and output resolutions increase, the induced kernels converge to a limiting kernel, in the sense of operator-norm convergence of their associated integral operators, allowing regularity assumptions to be stated independently of the encoding resolution. For regularized stochastic gradient descent, we establish upper bounds for decreasing and fixed step sizes, separating the encoding and regularization terms from optimization terms of order \(t^{-θ}\) and \(T^{-θ'}\), respectively, for any \(θ,θ'\in(0,1)\). We further prove lower bounds showing that these encoding-induced terms are generally unavoidable. The analysis is further extended to encoder--decoder neural networks through the limiting neural tangent kernel (NTK), yielding error bounds with an additional finite-width term and polynomial parameter and sample complexity guarantees when the encoding errors decay algebraically. The framework covers matrix-valued kernels constructed from radial and dot product kernels, NTKs arising from wide encoder--decoder neural networks, and encoder--decoder pairs based on Fourier, Legendre polynomial, wavelet, PCA, or pointwise sampling representations.