生成模型理论与方法

We study the recovery of forward and reverse quadratic optimal-transport maps from unpaired samples in high dimensions. We introduce a bidirectional neural framework in which the learned maps induce forward and backward displacement interpolants, while the training objective combines bidirectional quadratic action, discriminator-restricted Jensen-Shannon endpoint objectives, and two-sided cycle consistency. The construction requires neither precomputed sample pairings nor an explicit convex-potential parameterization. For absolutely continuous probability measures supported on a compact convex set, and under the stated generator-approximation, discriminator-richness, and minimizer-attainment conditions, we prove a population recovery theorem: for every prescribed accuracy, the sum of the corresponding \(L^2\) errors between any global minimizer and the forward and reverse quadratic Brenier maps is below that accuracy, provided the discriminator level is sufficiently large and the annealing action weight becomes sufficiently small. Moreover, the cycle loss is bounded above by \(λW_2^2(μ_0,μ_1)\). Complementary results quantify approximate invertibility and show that exact endpoint Jensen-Shannon divergence and cycle consistency control missing target mass and many-to-one map collapse, respectively. Experiments on Swiss roll, MNIST, CelebA, single-cell perturbation data, and chest X-ray images evaluate endpoint fidelity, transport cost, inverse consistency, and the geometry of the induced interpolations.

We study a class of distributionally robust optimization (DRO) problems for the statistical risk problem, formulated as minimax problems over the product of a Euclidean space and a Riemannian manifold. Because the resulting minimax landscape is nonconvex nonconcave in general, no globally convergent first order method is known to be available. We instead introduce the notion of a basin saddle point, a Nash equilibrium defined locally on the Cartesian product of a $δ$ basin around a connected component of the local minima critical set and a geodesic ball on the measure manifold. We develop an abstract convergence framework for a Riemannian gradient ascent multistep descent iteration to a basin saddle point under a local Łojasiewicz type growth condition, with exponent $β\in (1,2]$, in the $δ$ basin around connected components of the local minima critical sets. Under Lipschitz regularity of critical sets we establish linear convergence for $β= 2$ and polynomial convergence for $β\in (1,2)$ to a basin saddle point, with explicit dependence on the sectional curvature of the manifold. We then instantiate this framework for the statistical risk DRO problem over Gaussian measures, where the ambiguity set is naturally modeled as the product of Euclidean space and the Bures Wasserstein manifold of covariance matrices, which we relax to a penalized DRO formulation. We derive nonasymptotic Hessian estimates for the resulting Lagrangian, establish existence and local uniqueness of its maximizer, and prove that an alternating Riemannian gradient scheme converges to a basin saddle point of the penalized DRO problem, recovering the linear and polynomial rates of the abstract theory with all constants explicit in terms of data dimension, loss moments, and the reference covariance.

Continuous-latent Autoregressive Diffusion Transformer (AR-DiT) models have demonstrated immense potential in zero-shot speech generation. However, they still suffer from limited decoding stability when synthesizing long utterances or complex linguistic structures. This instability primarily stems from a restricted historical receptive field and an acoustic inertia dependency within the diffusion decoder, which causes the model to ignore semantic conditions. To address these challenges, we propose DiTAR+, a dual-optimization framework. First, we introduce Dilated Context Sampling to expand the macro-level historical receptive field without violating physical temporal continuity, thereby preventing cumulative error propagation. Second, we propose Hierarchical Acoustic Masking to prevent shallow layers from attending to acoustic pre-context, explicitly decoupling semantic alignment from acoustic detail reconstruction. Extensive experiments show that our framework effectively mitigates pronunciation errors and semantic hallucinations, enhances generation robustness on challenging sentences, and maintains exceptionally high speaker similarity throughout the entirety of long-form utterances. On the linguistically challenging ZH-Hard set, DiTAR+ reduces the word error rate from 12.478% to 9.893%, and on extended utterances of 25 to 35 seconds it improves speaker similarity from 0.741 to 0.759 while simultaneously lowering the word error rate from 2.778% to 2.173%, outperforming both discrete-token and pure flow-matching baselines.

Complex-valued signals, such as Magnetic Resonance Imaging (MRI) and audio spectrograms, are almost always modelled as flat two-channel Euclidean data. For nonzero values the amplitude-phase chart $z \mapsto (|z|, z/|z|)$ identifies the signal domain with the cylinder $(0, \infty) \times S^1$, on which we deliberately replace the inherited metric $dA^2 + A^2 dθ^2$ by the decoupled product metric $dA^2 + dθ^2$. In this empirical study we measure what that substitution costs and what it buys. By computing exact analytical bridges, we demonstrate that Cartesian paths induce a heavy-tailed distribution of angular velocity (power law index $\approx 1.0$), with nearly half of the probability paths exceeding an angular speed of $π$ under independent coupling, a rate no cylindrical path ever exceeds. To resolve this, we analyze Cylindrical Flow Matching (CyFM), which strictly bounds the regression target, and couple noise and data by exact minibatch Optimal Transport computed jointly over whole fields in the cylindrical metric. Although the transport-cost reduction of this coupling collapses with field dimension (from 86% for scalar pairs to 3% for $64\times64$ fields), its benefit to few-step generation does not: it lowers the few-step error of the cylindrical model by 3-60% at every evaluated resolution. With this coupling, CyFM has a lower error than the best Cartesian baseline at every step count up to $k = 8$ and every evaluated resolution, with all five seeds separated and without distillation, and at convergence we detect no significant difference between the two geometries. Finally, we expose the "Factorized Coupling Trap," showing that dimension-wise or patch-wise transport factorizations silently destroy the joint distribution of the data. All experiments are on synthetic complex fields.

We study loss-based filtering for finite-sum optimization with a subset of corrupted component functions whose gradients may be highly unreliable. Motivated by minimum-loss-based SGD (min-$k$-loss) and quantile-based methods for corrupted linear systems, we propose and analyze a general loss-filtering framework — Quantile-\(k\)-Loss SGD (Q\(k\)L-SGD) — that samples \(k\) component losses at each iteration and updates using an index chosen uniformly from the lower empirical \(q\)-quantile. We prove linear convergence of this family of methods under standard convexity assumptions, requiring the sample size to scale with the number of corruptions and a subset strong-convexity threshold. For the cases when large enough sampling is impossible or undesirable, we give a complementary small-sample probabilistic analysis that covers any sample size $k$ and the convergence behavior depends on the probability of selecting an outlier and on the curvature of the selected good step. Experiments on polynomial regression, regularized logistic regression, and regularized hinge loss show that intermediate quantiles often outperform both standard SGD and min-\(k\)-loss SGD. In particular, min-\(k\) often stalls by repeatedly selecting nearly solved components, while intermediate quantiles retain robustness and produce more informative updates.

Training diffusion models with multiple rewards requires distinguishing user preference from reward informativeness. User preference determines how much each reward should contribute to the overall objective; reward informativeness determines when its feedback is useful during denoising. Some rewards can meaningfully evaluate a sample as soon as global structure emerges, but others become informative only when the sample is nearly clean. To address both questions jointly, we propose ReCAST (Reward Credit ASsignment across T}imesteps), the first method, to our knowledge, for per-reward, timestep-dependent credit assignment in diffusion reward fine-tuning. ReCAST separates user preferences from temporal allocation through a reward-by-timestep weight matrix $W$, whose row sums match the user-specified reward budgets $λ$, while its column sums are equal, assigning the same total weight to each denoising step. Under these marginal constraints, ReCAST allocates weight according to each reward's informativeness, quantified by its Rényi discriminability gain at each step. These gains telescope to the total discriminability between the reward-induced positive policy and the current policy, providing a basis for temporal credit assignment. We evaluate ReCAST by training SD3.5-Medium under two distinct four-reward settings, each across five reward budgets $λ$. ReCAST improves the training rewards in one setting and matches them in the other, improves every held-out judge in both, and is preferred by an independent LLM-as-a-Judge. Together, these results show that ReCAST yields improvements that generalize beyond the training rewards and support its core principle: assigning each reward greater weight at the denoising timesteps where its feedback is most informative.

We study the problem of estimating unknown parameters in PDE-constrained inverse problems from noisy observations, where the PDE solution is approximated using Physics-Informed Neural Networks (PINNs). While PINNs have demonstrated remarkable empirical success, existing estimators often inherit the slow nonparametric convergence rate of the neural-network solution, leading to biased and statistically inefficient inference for the finite-dimensional parameters of interest. To address this, we propose a two-step debiased estimation procedure that combines neural-network-based nonparametric estimation with an influence-function-based bias correction. By eliminating the first-order sensitivity of the estimator to errors in the nuisance function, our procedure yields a $\sqrt{n}$-consistent and asymptotically normal estimator without requiring undersmoothing of the neural network component. We further extend this framework to Bayesian inference by replacing the original likelihood with a debiased quasi-likelihood and establish a Bernstein-von Mises theorem showing that the resulting posterior contracts at the $\sqrt{n}$-rate with an asymptotic covariance matching that of the frequentist estimator. As a by-product of our analysis, we establish near-minimax optimal convergence rates for estimating a nonparametric regression function and its derivatives in Sobolev spaces using neural networks. Extensive numerical experiments corroborate our theoretical findings and demonstrate the necessity of the proposed debiasing procedure for valid statistical inference in PDE-constrained inverse problems.

Many scientific applications require modeling how diffusive systems evolve over time, not merely their eventual steady states. While conventional steady-state analysis of partial differential equations (PDEs) on complex geometries is already hindered by costly volumetric meshing, transient analysis further requires sequential time stepping and careful step size selection. Grid-free Monte Carlo solvers such as walk on spheres (WoS) and walk on stars (WoSt) avoid this meshing bottleneck but remain largely limited to steady-state problems. We generalize WoS, for pure Dirichlet problems, and WoSt, for mixed Dirichlet--Neumann problems, to heat equations with initial conditions and time-dependent source and boundary data. We equip each random walk with a finite time budget and sample an exit time at every spatial step. If the exit time exceeds the remaining budget, the walk samples an interior point and evaluates the initial condition; otherwise, it continues with a reduced budget, accumulating source and boundary contributions. Our main technical contribution is a suite of kernel sampling and variance reduction techniques, including a low-bias, tabulation-free exit time sampler and efficient rejection samplers. Unlike grid-based transient solvers, our method directly estimates the solution at any requested time without volumetric meshing or sequential time marching. It also retains the parallel, progressive, and output-sensitive evaluation of WoS and WoSt while eliminating time step selection and temporal discretization bias entirely. Finally, we show how sharing walks enables efficient estimates at multiple target times.

Text-to-image (T2I) diffusion models inevitably internalize sensitive or non-compliant concepts from large-scale pretraining data, necessitating post-hoc concept erasure. However, existing erasure methods often lack explicit constraints on parameter updates, leading to over-intervention and unintended semantic drift. In addition, many methods rely on manually crafted counterfactual supervision, such as surrogate prompts, which incurs substantial data construction costs that limit scalability to new concepts. To address these limitations, we propose GRACE, a structured concept erasure framework designed to enable localized and selective intervention. Specifically, we introduce a semantically weighted sensitive subspace estimation to precisely lock intervention directions, and employ lightweight subspace-constrained adapters to prevent global semantic disturbance. To eliminate the dependency on manual prompt engineering, we design an automatically decoupled safe-anchor mechanism. To mitigate semantic drift induced by excessive intervention, we introduce an energy-driven dynamic gating mechanism that adaptively controls the timing and strength of intervention at inference. Extensive experiments demonstrate that our method achieves a superior balance between erasure effectiveness and generation fidelity. Compared with the average performance of five state-of-the-art (SOTA) concept erasure methods, our method improves the fine-grained NSFW reduction rate by $17.86%$, while reducing the macro-averaged target CLIP Score and preservation-oriented Fréchet Inception Distance (FID) by $4.75%$ and $50.58%$, respectively, indicating stronger concept suppression with substantially improved preservation of the original model's generative utility.

Full waveform inversion (FWI) estimates subsurface velocity from seismic recordings, but its ill-posedness and nonlinearity make accurate reconstruction strongly dependent on initialization and prior information. Diffusion posterior sampling provides a learned geological prior, yet directly coupling its denoiser to the nonlinear wave solver can yield unreliable physical guidance. We propose Physical-State-Guided Diffusion Sampling (PSG), which couples a persistent physical velocity to the diffusion prior through a Gaussian bridge. The physical state is refined by waveform fitting regularized by the denoised velocity, and in turn guides the reverse diffusion process. This formulation separates the wave-equation and denoiser gradients while preserving conventional FWI initialization and accumulated optimization history. On four OpenFWI families, PSG's terminal denoised estimates outperform classical and diffusion-based baselines under clean and missing-trace acquisitions and maintain strong structural recovery under measurement noise. Repeated stochastic runs preserve the dominant geological structures, with ensemble variability concentrated near geological interfaces and positively associated with local inversion error. A frozen OpenFWI-trained prior further supports inversion of the larger Marmousi, Overthrust, and BP2004 Salt models, recovering complex geological structures without retraining.

Humans organize knowledge into a taxonomy of concepts with nested levels of abstraction and a basic level at which people recognize and name objects with the least cognitive effort. Cobweb is a classic cognitive account of this ability, an incremental learner that builds a probabilistic concept hierarchy by maximizing category utility. We argue that diffusion models, although designed for image synthesis, implicitly perform the same computation. The noisy marginals of a diffusion model are Gaussian smoothings of the data distribution, and the modes of these marginals form a hierarchy that corresponds to a Cobweb tree of probabilistic prototypes in four respects. Both are hierarchical density models, both are hierarchical-Bayesian models with Gaussian prototypes, both treat categorization as score-following that reduces uncertainty, and in both a basic level emerges. We locate this basic level for a diffusion model at an intermediate noise level, where recent analyses show that the reverse process commits to the class identity of a sample. The two models differ mainly in how they represent and learn the taxonomy. Cobweb learns a discrete tree incrementally, whereas a diffusion model encodes a continuous, interpolable hierarchy in a single learned score field fit to the data distribution. We test the correspondence on MNIST and Fashion-MNIST by recovering the diffusion hierarchy through mode-finding and comparing the basic levels of the two models. This reframes diffusion as a cognitive model of concept formation and offers Cobweb a continuous, scalable instantiation.

Masked Diffusion Models (MDMs) have emerged as a compelling paradigm for language modeling, offering the capability for efficient parallel text generation. However, existing parallel sampling methods typically update multiple masked tokens independently and ignore the complex mutual dependencies among the masked tokens. This independent updating mechanism lacks global coordination and might lead to incoherent outputs. To address this limitation, we propose Representation-based Masked Diffusion Model (RMDM), a framework that leverages the text representation to explicitly encode global semantics and help to parallel update tokens more precisely. Specifically, we first encode text into a continuous semantic space using a pretrained encoder and learn an invertible transformation that normalizes the representation distribution to a Gaussian prior, facilitating efficient sampling during generation. Conditioned on this latent semantic representation, we train a masked diffusion model to learn the conditional text distribution, where the representation serves as global semantic guidance to coordinate parallel token updates and faithfully approximate the target distribution. Empirical results demonstrate that RMDM significantly improves generation quality, particularly in aggressive few-step sampling regimes.

Flow matching approaches to imaging inverse problems commonly incorporate measurements in two ways. Conditioning-based approaches supply measurement-derived information as a network input, often through concatenation, while inference-guided approaches combine an unconditional velocity field with a separate data-consistency update. In these common formulations, the forward model is not explicitly enforced within the learned conditional velocity field. We propose a principled parametrization of the measurement-conditional velocity field to solve inverse problems. Under linear interpolation, we express the conditional velocity $v(x_t,t,y)$ in terms of the posterior mean $E[x_1 | x_t,y]$, and characterize that mean as the unique minimizer of a variational objective whose data-consistency term is explicit. We further prove that the velocity field defines a probability flow from the source distribution to the measurement-conditioned posterior. Splitting the variational objective yields a conditional velocity parameterization with operator-dependent data-consistency updates, which we train end-to-end under the flow-matching objective, with no additional guidance at inference. Our method achieves state-of-the-art PSNR with $50\times$ fewer function evaluations than the strongest flow baseline. Varying the sampling steps provides test-time control over the distortion-perception trade-off without retraining.

Artificial intelligence increasingly relies on large-scale, distributed, and heterogeneous data collected by edge devices. However, the practice of edge intelligence remains challenging due to non-convex objectives, data heterogeneity, and complex wireless network topology. To address these issues, this paper proposes a consensus-based decentralized distributed swarm learning (CD-DSL) framework for wireless edge networks. Our CD-DSL integrates consensus optimization with particle swarm optimization (PSO), by reaching the model consensus among neighboring devices while leveraging the PSO exploration and exploitation. The consensus mechanism supports decentralized coordination without raw-data exchange, while PSO-inspired updates utilize historical and neighbor-shared experience to enhance exploration for non-convex optimization, improve robustness to data heterogeneity, and accelerate convergence. We further develop an adaptive neighbor-mixing strategy that learns performance-aware consensus weights, improving decentralized collaboration among heterogeneous edge devices. Theoretical analysis establishes that CD-DSL maintains participant consistency and achieves non-ergodic convergence to a neighborhood of a stationary point under non-convex objectives. Experimental results show that CD-DSL can mitigate the performance degeneration of existing decentralized baselines caused by heterogeneous data.

Existing evaluations of concept unlearning in text-to-image (T2I) diffusion models primarily rely on attack success rates obtained through automated adversarial prompt search. However, these metrics provide only empirical evidence over a finite set of queries and leave residual leakage over the broader prompt space largely unquantified. This limitation can lead to overestimating unlearning effectiveness and underestimating safety risks. To address this gap, we introduce a novel certification framework for T2I concept unlearning that provides high-confidence guarantees with bounded error on residual concept leakage. Our approach combines statistical certification with worst-case analysis along concept-relevant embedding directions to derive explicit upper bounds on leakage probability under user-specified confidence levels. We evaluate our framework across three major concept categories namely NSFW content, artistic styles, and celebrity identities, and six state-of-the-art unlearning methods. Certified leakage bounds consistently exceed standard attack success rates by 16.2%, uncovering substantial residual risks missed by existing evaluation protocols. Crucially, our results demonstrate that empirical attack-based evaluations can significantly underestimate residual leakage and establish certification as a necessary complement for reliable auditing of concept unlearning in T2I diffusion models.

Outliers are important for stress-testing algorithms and understanding system behaviour under rare conditions. Despite being commonly described as low-likelihood events, existing generative approaches rarely control likelihood explicitly. In this work, we introduce a measure-theoretic notion of outliers based on the distribution of log-likelihood values, which is guaranteed to assign higher probability mass to low-likelihood events with a specifiable magnitude. Building on this formulation, we derive how likelihood reweighting modifies the diffusion score and use this relation to motivate a controlled modification of the reverse-time dynamics. In particular, likelihood reweighting implies a scaling of the score function with a control term derived from the Radon-Nikodym derivative of the likelihood distributions. Correspondingly, the updated score function can be obtained with no retraining of the diffusion model. We exploit the Ornstein-Uhlenbeck semigroup underlying diffusion models to motivate an exponentially interpolated controller which approximates the true control. Experiments demonstrate controlled generation of low-likelihood samples while remaining consistent with the data geometry.

Diffusion and flow-matching schedules control the signal and noise coefficients that mix data and noise along affine probability paths. Minimizing a kinetic action defined on coefficient paths, motivated by optimal transport, helps explain strong baselines but remains model-agnostic and ignores prediction error. Here we introduce a model-aware schedule construction based on fiberwise optimal transport. At a fixed time and state on the probability path, compatible signal/noise decompositions form an affine fiber. We define a fiberwise prediction risk by averaging optimal-transport costs between the true and predictor-induced decompositions within these fibers. On a fixed coefficient curve, combining this risk with coefficient-path kinetic action yields a closed-form optimal time allocation. This construction extends to general linear prediction targets, and the risk profile can be estimated from an early baseline checkpoint. We evaluate DDPMs and flow matching across prediction targets, training configurations, risk-estimation checkpoints, datasets, and architectures. Our model-aware schedules consistently outperform strong baselines, including a 38.6% relative FID reduction for flow matching on CIFAR-10 at 16 function evaluations. Each model-agnostic kinetic baseline determines its own kinetic reference coordinate. In these coordinates, fiberwise-risk profiles from independently trained models in different settings align closely after normalization to unit area. The resulting schedule deformations used in training also align, suggesting empirical universality across the evaluated models and settings. Pretrained-checkpoint diagnostics extend this normalized-risk agreement to larger conditional latent diffusion and 2-RF models. A frozen analytic allocation template retains most of the model-aware improvement without further risk estimation or model-specific fitting.

Generative Marginalization Models (MaMs) have been recently introduced as efficient neural sampling models for any-order autoregressive modelling of discrete distributions. By learning both the marginal and conditional probabilities of a persistent-block Gibbs sampler, MaMs enable fast posterior evaluation with a single neural network forward pass. While prior work has considered MaMs to be distinct from Generative Flow Networks (GFlowNets), a well-established paradigm for inference in discrete stochastic models, we show that they are equivalent. Then, we also extend MaMs' sampling strategy to non-autoregressive generative processes. In particular, we describe an automatic criterion for full-state rejuvenation of the Gibbs sampler, derived from the Gelman-Rubin statistic, which plays a key role in speeding up learning convergence. Our experiments show that our method, called Particle GFlowNets, markedly accelerates training in large combinatorial spaces.

We introduce an amortized neural sampler that combines operator learning with flow methods for sampling. It maps SDE coefficient functions to pushforwards from a reference measure to the invariant measures, enabling efficient sampling across families of stochastic differential equations. Our framework shifts traditional sampling cost to an initial training phase, after which new SDE instances require only one encoder pass and a few ODE solver steps, independent of mixing time. To handle problems in high dimensions, we use Lagrangian trajectory sensors for the coefficient functions and cross attention in the architecture. We also theoretically establish the expressivity and resolution invariance of our framework. Experiments on 1D and 2D SDE families show competitive accuracy with substantial speedups over MCMC in regimes with slow mixing, transfer across sensor counts, and demonstration results on a 64D interacting particle SDE where traditional grid approaches are infeasible.

Maximum likelihood (ML) estimation is a principled and statistically efficient approach for learning probabilistic models. However, for unnormalized models, ML estimation requires evaluating the partition function and differentiating through it, which may not always be tractable. Score matching provides a practically viable alternative that circumvents this obstacle by fitting the score in a way that eliminates dependence on the normalizing constant. We derive the generalized score matching objective on a convex subset of $\mathbb{R}^{d}$ constructively starting from Minimum Probability Flow (MPF) learning, and show how classical score matching as well as domain-adapted variants for non-negative data arise naturally within the proposed framework. We show that the resulting objective is a {\it proper local scoring rule} of second-order, which provides the theoretical guarantee that the true density is recovered when the objective is minimized. Furthermore, for a model belonging to the exponential family, we establish convexity of the objective together with consistency of the finite-sample estimator under standard regularity conditions. Our derivation sheds new light on the scope and applicability of generalized score matching in various problem settings. We compare generalized score matching-based estimators on constrained domains, where the partition function is analytically intractable. We provide experimental results on parameter estimation for model densities belonging to the exponential family defined over convex subsets of $\mathbb{R}^{d}$, and a generative modeling use-case to demonstrate broader applicability of the proposed generalized score matching framework.

Diffusion language models decode tokens in parallel, but their bidirectional denoiser rules out the naive key--value (KV) cache behind fast autoregressive inference. Block diffusion restores caching by decoding block-by-block, and the block caches deployed on it so far are tied to attention: O(L)in memory and, if used as training-free retrofits, only an approximation of the model's computation. Both constraints can be overcome: sequence mixers that summarize finalized blocks into a reusable state support block caching, and the corresponding block-causal training objective makes the cache exact. We study this recipe at scale, pretraining three 3B block-diffusion denoisers (attention, mamba, and hybrid) on 300B tokens under one single-frontier objective and decoding all three through a single cached interface. Only the state-space cache is O(1) in sequence length: its memory and per-step latency stay constant at any context length, while an attention cache remains O(L). At 256k tokens (where attention has grown to 82GB and 29 ms/step), the Mamba cache delivers 4.3x lower latency, 11x less memory, and 2.6x higher single-stream throughput; and because that footprint is constant it scales with batch as well, reaching 14x the aggregate throughput, where attention cannot run beyond a single stream. The same linear-state bias lets the Mamba and hybrid backbones keep retrieving out to 8-16x their training length, whereas attention's retrieval collapses at 2x, at no measured quality cost.

Natural load-bearing patterns such as leaf venation, trabecular bone, and spider webs achieve high stiffness per unit mass, yet classical topology optimizers rarely reach such geometries, and few let engineers express structural design intent through natural language. This work treats a frozen text-to-image diffusion model as a training-free source of design knowledge and distills it into the physics loop of density-based topology optimization via score distillation sampling, so that a text prompt becomes an explicit, machine-interpretable representation of engineer intent. The prompt-induced generative gradient and the finite element sensitivity are combined at every iteration, letting physics decide which prompt-induced features survive. In 245 primary SDS runs spanning four geometric domains and two physics regimes, 38 of 49 prompt--domain combinations achieved statistically significant compliance reductions (up to $-31.5%$ mechanical and $-23.0%$ thermoelastic), outperforming gradient-based baselines. Cross-domain morphological analysis identifies a recurring structural signature of improvement: in most domains the generative prior suppresses dead-end branches in the rib skeleton, with endpoint--compliance correlation $r = +0.56$ to $+0.99$. A Heaviside projection with $β$-continuation resolves a pronounced intermediate-density tendency in this diffusion--physics coupling ($42.6%$ to $<3%$), and an automated skeleton-based pipeline converts optimized density fields into \rev{candidate geometry ready for computer-aided design. By retargeting the generative prior across domains, loading conditions, and physics objectives through a change of text prompt, with each new problem's physics setup specified separately, the framework uses a pretrained generative model as a reusable, training-free prior for engineering design.

Time series foundation models (TSFMs) are increasingly pre-trained on synthetically generated time series trajectories, where the data generating process is known. Current pre-training recipes are based on loss objectives which compare TSFM outputs to realized future values of each trajectory. We instead propose loss objectives which compare TSFM outputs to the conditional forecast distribution of each trajectory, a procedure we call synthetic data distillation (SDD). SDD corresponds to a Rao-Blackwellization of the training objective, in that it leaves the expectation of stochastic gradients unchanged while provably reducing the covariance of the stochastic gradient under the Loewner partial ordering. We empirically validate SDD on a TSFM model family of sizes from $4$M to $2.5$B parameters, and observe faster convergence of validation loss at every model size: on Gaussian Process data, SDD attains or improves upon the Status Quo loss whilst requiring $10%-40%$ less training iterations.

Image-to-3D models can generate visually compelling 3D assets from a single RGB image, but their geometry is often only loosely constrained by the available observations, limiting their use in applications that require geometric fidelity. In many real-world settings, however, partial geometric observations of the object may be available at test time. We introduce a training-free framework for incorporating such evidence into pretrained image-to-3D generative models without retraining or finetuning. To do this, we guide generation using a ray-consistent observation likelihood defined over the model's occupancy representation, combining surface occupancy and free-space evidence. Applied to SAM 3D and its multi-view extension, our approach substantially improves geometric fidelity across different levels of observability, as well as visual quality. Our results demonstrate that pretrained image-to-3D models can effectively integrate partial geometric observations through explicit test-time guidance, complementing their learned generative priors without modifying the underlying model.

Robot actions are temporally correlated trajectories whose frequency components encode motion at different scales with highly non-uniform energy distributions. Yet Flow Matching--based vision-language-action (VLA) models typically generate actions in temporal coordinates, without explicitly modeling or systematically leveraging this frequency heterogeneity. We introduce FreqFM, a frequency-conditioned Flow Matching framework for VLA models. It raises action frequency from an implicit trajectory property to an explicit conditioning dimension that spans the entire generation pipeline. Concretely, in DCT frequency coordinates, FreqFM constructs a spectrum-matched source distribution, adaptively balances the objective across frequencies, and constrains per-frequency guidance residuals using the corresponding reference transport scales. FreqFM integrates into existing Flow Matching action experts without changing the VLA backbone. Across LIBERO, LIBERO-Plus, and VLA-Arena, FreqFM consistently improves performance, including a 9.3-point gain on LIBERO-Plus, and further demonstrates its effectiveness on six real-robot tasks.

Multi-view diffusion models have shown strong performance in scenes with strong geometric priors and sparse semantics, such as indoor rooms or simple outdoor environments (e.g., fields, courtyards). However, they often fail to maintain cross-view consistency under camera rotation, especially in structurally complex urban environments. Without explicit modeling of spherical correspondence across views, existing approaches tend to produce object duplication, structural distortion, and layout inconsistency. To address this limitation, we propose StreetDiff, a multi-view diffusion framework that explicitly enforces cross-view alignment during denoising. StreetDiff introduces a Panorama--Perspective Synergy design to decouple global layout reasoning from local detail synthesis, and incorporates a Panorama Alignment Module (PAM) that establishes spherical-projection-based attention constraints across views. By injecting structured alignment constraints without modifying the diffusion backbone, our framework achieves robust cross-view coherence in challenging urban street scene generation tasks. In addition, we construct Street360, a large-scale HDR multi-view urban panorama dataset. Extensive experiments demonstrate that StreetDiff significantly improves structural consistency and visual fidelity compared to prior multi-view diffusion generation methods.

Although text-to-image diffusion models generally exhibit strong prompt-following ability, we identify a persistent and previously underexplored failure pattern in which a small subset of prompts differing only in the object consistently fails to realize the same target concept under identical generation settings. We term this phenomenon object-dependent concept brittleness. Such cases suggest systematic internal blind spots rather than random sampling noise. In this paper, we present an interpretability-oriented framework to audit and minimally correct these failures. Our key idea is to analyze denoising trajectories in a step-wise sparse autoencoder (SAE) space, where abstract style and attribute concepts become more separable than in the raw denoising representation. This sparse space enables us to compare successful and failed generations, identify concept dimensions whose evidence is missing, weakened, or temporally delayed, and construct class-level concept prototypes from reliable class-consistent samples. Based on this audit process, we introduce a lightweight inference-time correction strategy that interpolates denoising features toward the corresponding prototype in SAE space. Rather than serving as a task-specific retraining method, this intervention acts as a validation of the diagnosed concept deficiency. We evaluate the proposed framework on style and attribute failure cases across multiple diffusion backbones, with significant improvements in concept consistency, text fidelity, and repair success. Further analyses show that deeper denoising representations provide clearer concept structure, while early-stage intervention offers the strongest correction leverage. Code is available at https://github.com/Metecade/Object-Dependent-Concept-Brittleness.

Crystal structure prediction (CSP) is fundamental to computational materials discovery. Generative models including CDVAE, DiffCSP, FlowMM, and CrystalFlow learn stable-crystal distributions directly, but diffusion and flow-matching inference requires tens to thousands of sequential network evaluations per candidate. We introduce uFlowCSP, a MeanFlow-based CSP model that learns the average, rather than instantaneous, probability-flow velocity. It generates a complete structure in one to five evaluations, delivering 5x-58x faster inference with equal or better performance. A chemistry- and symmetry-aware Transformer uses canonical atom ordering, global composition, and per-token chemistry embeddings. A coarse crystal-system token is used only during training; it provides additive gains, particularly improving space-group agreement despite being absent at inference, which remains formula-only. On MP-20 with 20 candidates per target, one step matches CrystalFlow (78.38% vs. 78.34%) with 100x fewer evaluations and about 10x lower wall-clock time. Five steps reach 83.64%, exceeding CrystalFlow (78.34% at 2,000 evaluations) and DiffCSP (77.93% at about 20,000), while using 20x fewer evaluations. uFlowCSP generates 10,000 structures in 0.39-1.31 minutes, versus 6.5 for CrystalFlow and 76.1 for DiffCSP. Under CSPBench's energy-ranked top-five structure-and-space-group criterion, five-step uFlowCSP reaches 72%/72%/65% structure, space-group, and consensus match rates. CrystalFlow reaches 78%/73%/68% at 100 steps but falls to 49%/32%/31% at five. Thus, uFlowCSP improves accuracy per network evaluation, not merely peak accuracy.

Diffusion models have achieved remarkable success in generative tasks across various areas, however their training process raises significant privacy concerns, particularly under membership inference attacks (MIAs). Prior studies on privacy-preserving of diffusion models fail to balance privacy, utility, and efficiency. To address this gap, we propose a novel framework of privacy-preserving diffusion models, Adaptive Diffusion Freezing (ADF), which can defend against MIAs with better trade-off. By leveraging cross-timestep adaptive freezing training, ADF explicitly control the participation of different data subsets across diffusion timesteps via a mask matrix, which reduces the over-memorization and leads to more uniform model behaviors between member and nonmember samples. To construct a freezing mask matrix that effectively reduce membership leakage without unnecessarily harming generation quality, we introduce a pretraining-based risk-aware freezing policy to estimate MIA risk based on memorization tendency, and suppress the contribution of the subset-timestep pairs with higher risk. Evaluations on multiple datasets demonstrate that ADF provides effective defense performance as well as state-of-the-art privacy-utility-efficiency trade-off performance compared to various baselines.

Minimum-norm confidence envelope strategies offer a nonparametric approach to constructing nonasymptotic, simultaneous confidence regions for band-limited functions, exploiting the theory of Reproducing Kernel Hilbert Spaces (RKHS). While the finite-sample coverage guarantees of these envelopes have been established, their consistency has not been analyzed so far. In this paper, we study this construction, here termed the Minimum-Norm Confidence Envelope (MiNCE) framework, and establish the strong uniform consistency of the resulting bands, both for noise-free and noisy observation models, under mild assumptions on the measurement noises. We further extend this formulation to the frequency domain, deriving nonasymptotic, simultaneous, strongly uniformly consistent confidence bands for the smoothed spectra. Numerical experiments in nonparametric regression and spectral estimation empirically confirm our theoretical results, illustrating the contraction of the confidence envelopes toward the target function as the sample size increases.