Generative Modeling

Generative models are increasingly deployed as substitutes for real data in downstream scientific workflows, yet standard evaluation criteria remain focused on marginal distribution matching. We argue that this represents a fundamental gap: downstream inference is rarely a marginal operation, and a model that passes every univariate diagnostic can still produce structurally unreliable synthetic data. We introduce covariance-level dependence fidelity, measured by D_Sigma(P,Q) = ||Sigma_P - Sigma_Q||_F, as a principled, computable criterion for evaluating whether a generative model preserves the joint structure of data beyond its univariate marginals. Three results formalise this criterion. First, marginal fidelity provides no constraint on dependence structure: D_Sigma can be made arbitrarily large while all univariate marginals match exactly. Second, covariance divergence induces quantifiable downstream instability, including sign reversals in population regression coefficients. Third, bounding D_Sigma provides positive stability guarantees for dependence-sensitive procedures such as PCA via Davis-Kahan-type bounds. Empirical validation across three domains, image data (Fashion-MNIST VAE, n = 60,000), bulk RNA-seq (TCGA-BRCA, n = 1,111), and a small-sample stress test (Alzheimer's gene expression, n = 113), shows that D_Sigma/delta consistently distinguishes structure-discarding from structure-preserving generators in cases where standard marginal diagnostics show little separation, confirming that covariance-level fidelity provides information orthogonal to existing evaluation metrics across domains and sample sizes.

A single egocentric image typically captures only a small portion of the floor, yet a complete metric traversability map of the surroundings would better serve applications such as indoor navigation. We introduce FlatLands, a dataset and benchmark for single-view bird's-eye view (BEV) floor completion. The dataset contains 270,575 observations from 17,656 real metric indoor scenes drawn from six existing datasets, with aligned observation, visibility, validity, and ground-truth BEV maps, and the benchmark includes both in- and out-of-distribution evaluation protocols. We compare training-free approaches, deterministic models, ensembles, and stochastic generative models. Finally, we instantiate the task as an end-to-end monocular RGB-to-floormaps pipeline. FlatLands provides a rigorous testbed for uncertainty-aware indoor mapping and generative completion for embodied navigation.

Diffusion-based trajectory planners can synthesize rich, multimodal action sequences for offline reinforcement learning, but their iterative denoising incurs substantial inference-time cost, making closed-loop planning slow under tight compute budgets. We study the problem of achieving diffusion-like trajectory planning behavior with one-step inference, while retaining the ability to sample diverse candidate plans and condition on the current state in a receding-horizon control loop. Our key observation is that conditional trajectory generation fails under naïve distribution-matching objectives when the similarity measure used to align generated trajectories with the dataset is dominated by unconstrained future dimensions. In practice, this causes attraction toward average trajectories, collapses action diversity, and yields near-static behavior. Our key insight is that conditional generative planning requires a conditioning-aware notion of neighborhood: trajectory updates should be computed using distances in a compact key space that reflects the condition, while still applying updates in the full trajectory space. Building on this, we introduce Keyed Drifting Policies (KDP), a one-step trajectory generator trained with a drift-field objective that attracts generated trajectories toward condition-matched dataset windows and repels them from nearby generated samples, using a stop-gradient drifted target to amortize iterative refinement into training. At inference, the resulting policy produces a full trajectory window in a single forward pass. Across standard RL benchmarks and real-time hardware deployments, KDP achieves strong performance with one-step inference and substantially lower planning latency than diffusion sampling. Project website, code and videos: https://keyed-drifting.github.io/

Generative models based on the Flow Matching objective, particularly Rectified Flow, have emerged as a dominant paradigm for efficient, high-fidelity image synthesis. However, while existing research heavily prioritizes generation quality and architectural scaling, the underlying dynamics of how RF models memorize training data remain largely underexplored. In this paper, we systematically investigate the memorization behaviors of RF through the test statistics of Membership Inference Attacks (MIA). We progressively formulate three test statistics, culminating in a complexity-calibrated metric that successfully decouples intrinsic image spatial complexity from genuine memorization signals. This calibration yields a significant performance surge -- boosting attack AUC by up to 15% and the privacy-critical TPR@1%FPR metric by up to 45% -- establishing the first non-trivial MIA specifically tailored for RF. Leveraging these refined metrics, we uncover a distinct temporal pattern: under standard uniform temporal training, a model's susceptibility to MIA strictly peaks at the integration midpoint, a phenomenon we justify via the network's forced deviation from linear approximations. Finally, we demonstrate that substituting uniform timestep sampling with a Symmetric Exponential (U-shaped) distribution effectively minimizes exposure to vulnerable intermediate timesteps. Extensive evaluations across three datasets confirm that this temporal regularization suppresses memorization while preserving generative fidelity.

Federated learning has emerged as a paradigm to train models collaboratively on inherently distributed client data while safeguarding privacy. In this context, personalized federated learning tackles the challenge of data heterogeneity by equipping each client with a dedicated model. A prevalent strategy decouples the model into a shared feature extractor and a personalized classifier head, where the latter actively guides the representation learning. However, previous works have focused on classifier head-guided personalization, neglecting the potential personalized characteristics in the representation distribution. Building on this insight, we propose pFedGM, a method based on Gaussian generative modeling. The approach begins by training a Gaussian generator that models client heterogeneity via weighted re-sampling. A balance between global collaboration and personalization is then struck by employing a dual objective: a shared objective that maximizes inter-class distance across clients, and a local objective that minimizes intra-class distance within them. To achieve this, we decouple the conventional Gaussian classifier into a navigator for global optimization, and a statistic extractor for capturing distributional statistics. Inspired by the Kalman gain, the algorithm then employs a dual-scale fusion framework at global and local levels to equip each client with a personalized classifier head. In this framework, we model the global representation distribution as a prior and the client-specific data as the likelihood, enabling Bayesian inference for class probability estimation. The evaluation covers a comprehensive range of scenarios: heterogeneity in class counts, environmental corruption, and multiple benchmark datasets and configurations. pFedGM achieves superior or competitive performance compared to state-of-the-art methods.

We introduce COT-FM, a general framework that reshapes the probability path in Flow Matching (FM) to achieve faster and more reliable generation. FM models often produce curved trajectories due to random or batchwise couplings, which increase discretization error and reduce sample quality. COT-FM fixes this by clustering target samples and assigning each cluster a dedicated source distribution obtained by reversing pretrained FM models. This divide-and-conquer strategy yields more accurate local transport and significantly straighter vector fields, all without changing the model architecture. As a plug-and-play approach, COT-FM consistently accelerates sampling and improves generation quality across 2D datasets, image generation benchmarks, and robotic manipulation tasks.

Flow Matching enables simulation-free training of generative models on Riemannian manifolds, yet sampling typically still relies on numerically integrating a probability-flow ODE. We propose Riemannian MeanFlow (RMF), extending MeanFlow to manifold-valued generation where velocities lie in location-dependent tangent spaces. RMF defines an average-velocity field via parallel transport and derives a Riemannian MeanFlow identity that links average and instantaneous velocities for intrinsic supervision. We make this identity practical in a log-map tangent representation, avoiding trajectory simulation and heavy geometric computations. For stable optimization, we decompose the RMF objective into two terms and apply conflict-aware multi-task learning to mitigate gradient interference. RMF also supports conditional generation via classifier-free guidance. Experiments on spheres, tori, SO(3), and SE(3) demonstrate competitive one-step sampling with improved quality-efficiency trade-offs and substantially reduced sampling cost.

We reveal a precise mathematical framework about a new family of generative models which we call Gradient Flow Drifting. With this framework, we prove an equivalence between the recently proposed Drifting Model and the Wasserstein gradient flow of the forward KL divergence under kernel density estimation (KDE) approximation. Specifically, we prove that the drifting field of drifting model (arXiv:2602.04770) equals, up to a bandwidth-squared scaling factor, the difference of KDE log-density gradients $\nabla \log p_{\mathrm{kde}} - \nabla \log q_{\mathrm{kde}}$, which is exactly the particle velocity field of the Wasserstein-2 gradient flow of $KL(q\|p)$ with KDE-approximated densities. Besides that, this broad family of generative models can also include MMD-based generators, which arises as special cases of Wasserstein gradient flows of different divergences under KDE approximation. We provide a concise identifiability proof, and a theoretically grounded mixed-divergence strategy. We combine reverse KL and $χ^2$ divergence gradient flows to simultaneously avoid mode collapse and mode blurring, and extend this method onto Riemannian manifold which loosens the constraints on the kernel function, and makes this method more suitable for the semantic space. Preliminary experiments on synthetic benchmarks validate the framework.

Learning to sample from complex unnormalized distributions is a fundamental challenge in computational physics and machine learning. While score-based and variational methods have achieved success in continuous domains, extending them to discrete or mixed-variable systems remains difficult due to ill-defined gradients or high variance in estimators. We propose a unified, target-gradient-free generative sampling framework applicable across diverse state spaces. Building on the fact that detailed balance implies the time-reversibility of the equilibrium stochastic process, we enforce this symmetry as a statistical constraint. Specifically, using a prescribed physical transition kernel (such as Metropolis-Hastings), we minimize the Maximum Mean Discrepancy (MMD) between the joint distributions of forward and backward Markov trajectories. Crucially, this training procedure relies solely on energy evaluations via acceptance ratios, circumventing the need for target score functions or continuous relaxations. We demonstrate the versatility of our method on three distinct benchmarks: (1) a continuous multi-modal Gaussian mixture, (2) the discrete high-dimensional Ising model, and (3) a challenging hybrid system coupling discrete indices with continuous dynamics. Experiments show that our framework accurately reproduces thermodynamic observables and captures mode-switching behavior across all regimes, offering a physically grounded and universally applicable alternative for equilibrium sampling.

Financial time-series forecasting is a high-stakes problem where regime shifts and shocks make point-accurate yet overconfident models dangerous. We propose Uncertainty-Gated Generative Modeling (UGGM), which treats uncertainty as an internal control signal that gates (i) representation via gated reparameterization, (ii) propagation via similarity and confidence routing, and (iii) generation via uncertainty-controlled predictive distributions, together with uncertainty-driven regularization and calibration to curb miscalibration. Instantiated on Weak Innovation AutoEncoder (WIAE-GPF), our UG-WIAE-GPF significantly improves risk-sensitive forecasting, delivering a 63.5% MSE reduction on NYISO (0.3508 $\rightarrow$ 0.1281), with improved robustness under shock intervals (mSE: 0.2739 $\rightarrow$ 0.1748).

Drifting models train one-step generators by optimizing a kernel-induced mean-shift discrepancy between the data and model distributions, with Laplace kernels used by default in practice. At each point, this discrepancy compares the kernel-weighted displacement toward nearby data samples with the corresponding displacement toward nearby model samples, thereby defining a transport direction for generated samples. In this paper, we show that drifting is more closely connected to score-based generative modeling than it may first appear, establishing a precise link to the score-matching principle underlying diffusion models. For Gaussian kernels, the population mean-shift field exactly equals the difference between the scores (i.e., the gradient-log-densities) of the Gaussian-smoothed data and model distributions. This identity follows from Tweedie's formula, which links the score of a Gaussian-smoothed density to its conditional mean, and implies that Gaussian-kernel drifting is exactly a score-matching objective on smoothed distributions. More generally, we derive an exact decomposition for radial kernels in which mean shift equals a score-based field plus a residual term. For the practical Laplace kernel, we further show theoretically and empirically that this residual is negligible in high dimension, implying that the transport field used in practice is nearly score-based. Our results reveal a structural connection to diffusion models: both methods use score-mismatch transport directions, but drifting realizes the score nonparametrically through kernel-based estimates, whereas diffusion models learn it parametrically with neural networks.

Latent generative models have emerged as powerful priors for solving inverse problems. These models typically represent a class of natural signals at a single, fixed complexity, governed by the latent dimensionality. This can be limiting: depending on the problem, a latent dimensionality that is too small may result in high representation error, while one that is too large may overfit to noise. We develop tunable latent priors for diffusion models, normalizing flows, and variational autoencoders, leveraging nested dropout. Across tasks including compressed sensing, inpainting, denoising, and phase retrieval, we show empirically that tunable priors consistently achieve lower reconstruction errors than fixed-complexity baselines. In the linear denoising setting, we derive the optimal complexity in closed form, showing how it depends on the noise level and the signal spectrum. This work demonstrates the potential of tunable latent generative priors and motivates both the development of supporting theory and their application across a wide range of inverse problems.

Constrained generative modeling is fundamental to applications such as robotic control and autonomous driving, where models must respect physical laws and safety-critical constraints. In real-world settings, these constraints rarely take the form of simple linear inequalities, but instead complex feasible regions that resemble road maps or other structured spatial domains. We propose a constrained generation framework that generates samples directly within such feasible regions while preserving realism. Our method fine-tunes a pretrained generative model to enforce constraints while maintaining generative fidelity. Experimentally, our method exhibits characteristics distinct from existing fine-tuning and training-free constrained baselines, revealing a new compromise between constraint satisfaction and sampling quality.

The emergence of generative AI models has dramatically expanded the availability and use of synthetic data across scientific, industrial, and policy domains. While these developments open new possibilities for data analysis, they also raise fundamental statistical questions about when synthetic data can be used in a valid, reliable, and principled manner. This paper reviews the current landscape of synthetic data generation and use from a statistical perspective, with the goal of clarifying the assumptions under which synthetic data can meaningfully support downstream discovery, inference, and prediction. We survey major classes of modern generative models, their intended use cases, and the benefits they offer, while also highlighting their limitations and characteristic failure modes. We additionally examine common pitfalls that arise when synthetic data are treated as surrogates for real observations, including biases from model misspecification, attenuated uncertainty, and difficulties in generalization. Building on these insights, we discuss emerging frameworks for the principled use of synthetic data. We conclude with practical recommendations, open problems, and cautions intended to guide both method developers and applied researchers.

To explore the feasibility of avoiding the confident error (or hallucination) of generation models (GMs), we formalise the system of GMs as a class of stochastic dynamical systems through the lens of control theory. Numerous factors can be attributed to the hallucination of the learning process of GMs, utilising knowledge of control theory allows us to analyse their system functions and system responses. Due to the high complexity of GMs when using various optimization methods, we cannot figure out their solution of Laplace transform, but from a macroscopic perspective, simulating the source response provides a virtual way to address the hallucination of GMs. We also find that the training progress is consistent with the corresponding system response, which offers us a useful way to develop a better optimization component. Finally, the hallucination problem of GMs is fundamentally optimized by using Laplace transform analysis.

Many modern biological assays, including RNA sequencing, yield integer-valued counts that reflect the number of molecules detected. These measurements are often not at the desired resolution: while the unit of interest is typically a single cell, many measurement technologies produce counts aggregated over sets of cells. Although recent generative frameworks such as diffusion and flow matching have been extended to non-Euclidean and discrete settings, it remains unclear how best to model integer-valued data or how to systematically deconvolve aggregated observations. We introduce Count Bridges, a stochastic bridge process on the integers that provides an exact, tractable analogue of diffusion-style models for count data, with closed-form conditionals for efficient training and sampling. We extend this framework to enable direct training from aggregated measurements via an Expectation-Maximization-style approach that treats unit-level counts as latent variables. We demonstrate state-of-the-art performance on integer distribution matching benchmarks, comparing against flow matching and discrete flow matching baselines across various metrics. We then apply Count Bridges to two large-scale problems in biology: modeling single-cell gene expression data at the nucleotide resolution, with applications to deconvolving bulk RNA-seq, and resolving multicellular spatial transcriptomic spots into single-cell count profiles. Our methods offer a principled foundation for generative modeling and deconvolution of biological count data across scales and modalities.

A major focus in designing methods for learning distributions defined on manifolds is to alleviate the need to implicitly learn the manifold so that learning can concentrate on the data distribution within the manifold. However, accomplishing this often leads to compute-intensive solutions. In this work, we propose a simple modification to denoising score-matching in the ambient space to implicitly account for the manifold, thereby reducing the burden of learning the manifold while maintaining computational efficiency. Specifically, we propose a simple decomposition of the score function into a known component $s^{base}$ and a remainder component $s-s^{base}$ (the learning target), with the former implicitly including information on where the data manifold resides. We derive known components $s^{base}$ in analytical form for several important cases, including distributions over rotation matrices and discrete distributions, and use them to demonstrate the utility of this approach in those cases.

Generative models have recently advanced $\textit{de novo}$ protein design by learning the statistical regularities of natural structures. However, current approaches face three key limitations: (1) Existing methods cannot jointly learn protein geometry and design tasks, where pretraining can be a solution; (2) Current pretraining methods mostly rely on local, non-rigid atomic representations for property prediction downstream tasks, limiting global geometric understanding for protein generation tasks; and (3) Existing approaches have yet to effectively model the rich dynamic and conformational information of protein structures. To overcome these issues, we introduce $\textbf{RigidSSL}$ ($\textit{Rigidity-Aware Self-Supervised Learning}$), a geometric pretraining framework that front-loads geometry learning prior to generative finetuning. Phase I (RigidSSL-Perturb) learns geometric priors from 432K structures from the AlphaFold Protein Structure Database with simulated perturbations. Phase II (RigidSSL-MD) refines these representations on 1.3K molecular dynamics trajectories to capture physically realistic transitions. Underpinning both phases is a bi-directional, rigidity-aware flow matching objective that jointly optimizes translational and rotational dynamics to maximize mutual information between conformations. Empirically, RigidSSL variants improve designability by up to 43% while enhancing novelty and diversity in unconditional generation. Furthermore, RigidSSL-Perturb improves the success rate by 5.8% in zero-shot motif scaffolding and RigidSSL-MD captures more biophysically realistic conformational ensembles in G protein-coupled receptor modeling.

Many creative multimedia systems are built upon visual primitives such as strokes or textures, which are difficult to collect at scale and fundamentally different from natural image data. This data scarcity makes it challenging for modern generative models to learn expressive and controllable primitives, limiting their use in process-aware content creation. In this work, we study the problem of learning human-like brushstroke generation from a small set of hand-drawn samples (n=470) and propose StrokeDiff, a diffusion-based framework with Smooth Regularization (SmR). SmR injects stochastic visual priors during training, providing a simple mechanism to stabilize diffusion models under sparse supervision without altering the inference process. We further show how the learned primitives can be made controllable through a Bézier-based conditioning module and integrated into a complete stroke-based painting pipeline, including prediction, generation, ordering, and compositing. This demonstrates how data-efficient primitive modeling can support expressive and structured multimedia content creation. Experiments indicate that the proposed approach produces diverse and structurally coherent brushstrokes and enables paintings with richer texture and layering, validated by both automatic metrics and human evaluation.

Score-based generative models (SGMs) have achieved remarkable empirical success, motivating their application to a broad range of data distributions. However, extending them to heavy-tailed targets remains a largely open problem. Although dedicated models for heavy-tailed distributions have been proposed, their generative fidelity remains unclear and they lack solid theoretical foundations, leaving important questions open in this regime. In this paper, we address this gap through two theoretical contributions. First, we show that combining early stopping with a suitable initialization is sufficient to extend the diffusion framework to any target distribution; in particular, we establish the well-posedness of the backward process and prove convergence of the approximated diffusion in KL divergence. Second, we derive novel theoretical guarantees for generation with normalizing flows, obtaining convergence results that hold under mild conditions on the flow family and without any assumption on the tail behavior of the target distribution. Building on these results, we propose a unified generative framework for heavy-tailed distributions: a normalizing flow is first trained to capture the tail behavior and is then used as an initialization prior for an SGM, which refines the samples by recovering fine-grained structural details. This design leverages the complementary strengths of the two model classes within a theoretically principled pipeline, overcoming the limitations of existing approaches.

This paper introduces General Proximal Flow Networks (GPFNs), a generalization of Bayesian Flow Networks that broadens the class of admissible belief-update operators. In Bayesian Flow Networks, each update step is a Bayesian posterior update, which is equivalent to a proximal step with respect to the Kullback-Leibler divergence. GPFNs replace this fixed choice with an arbitrary divergence or distance function, such as the Wasserstein distance, yielding a unified proximal-operator framework for iterative generative modeling. The corresponding training and sampling procedures are derived, establishing a formal link to proximal optimization and recovering the standard BFN update as a special case. Empirical evaluations confirm that adapting the divergence to the underlying data geometry yields measurable improvements in generation quality, highlighting the practical benefits of this broader framework.

Foundation models have transformed domains from language to genomics by learning general-purpose representations from large-scale, heterogeneous data. We introduce TradeFM, a 524M-parameter generative Transformer that brings this paradigm to market microstructure, learning directly from billions of trade events across >9K equities. To enable cross-asset generalization, we develop scale-invariant features and a universal tokenization scheme that map the heterogeneous, multi-modal event stream of order flow into a unified discrete sequence -- eliminating asset-specific calibration. Integrated with a deterministic market simulator, TradeFM-generated rollouts reproduce key stylized facts of financial returns, including heavy tails, volatility clustering, and absence of return autocorrelation. Quantitatively, TradeFM achieves 2-3x lower distributional error than Compound Hawkes baselines and generalizes zero-shot to geographically out-of-distribution APAC markets with moderate perplexity degradation. Together, these results suggest that scale-invariant trade representations capture transferable structure in market microstructure, opening a path toward synthetic data generation, stress testing, and learning-based trading agents.

Flow matching has emerged as a simulation-free alternative to diffusion-based generative modeling, producing samples by solving an ODE whose time-dependent velocity field is learned along an interpolation between a simple source distribution (e.g., a standard normal) and a target data distribution. Flow-based methods often exhibit greater training stability and have achieved strong empirical performance in high-dimensional settings where data concentrate near a low-dimensional manifold, such as text-to-image synthesis, video generation, and molecular structure generation. Despite this success, existing theoretical analyses of flow matching assume target distributions with smooth, full-dimensional densities, leaving its effectiveness in manifold-supported settings largely unexplained. To this end, we theoretically analyze flow matching with linear interpolation when the target distribution is supported on a smooth manifold. We establish a non-asymptotic convergence guarantee for the learned velocity field, and then propagate this estimation error through the ODE to obtain statistical consistency of the implicit density estimator induced by the flow-matching objective. The resulting convergence rate is near minimax-optimal, depends only on the intrinsic dimension, and reflects the smoothness of both the manifold and the target distribution. Together, these results provide a principled explanation for how flow matching adapts to intrinsic data geometry and circumvents the curse of dimensionality.

Egocentric manipulation videos are highly challenging due to severe occlusions during interactions and frequent object entries and exits from the camera view as the person moves. Current methods typically focus on recovering either hand or object pose in isolation, but both struggle during interactions and fail to handle out-of-sight cases. Moreover, their independent predictions often lead to inconsistent hand-object relations. We introduce WHOLE, a method that holistically reconstructs hand and object motion in world space from egocentric videos given object templates. Our key insight is to learn a generative prior over hand-object motion to jointly reason about their interactions. At test time, the pretrained prior is guided to generate trajectories that conform to the video observations. This joint generative reconstruction substantially outperforms approaches that process hands and objects separately followed by post-processing. WHOLE achieves state-of-the-art performance on hand motion estimation, 6D object pose estimation, and their relative interaction reconstruction. Project website: https://judyye.github.io/whole-www

Generative Flow Networks (GFlowNets) learn to sample diverse candidates in proportion to a reward function, making them well-suited for scientific discovery, where exploring multiple promising solutions is crucial. Further extending GFlowNets to multi-objective settings has attracted growing interest as real-world applications often involve multiple, conflicting objectives. However, existing approaches require joint training for each combination of objectives, meaning that any change in the objective set necessitates retraining from scratch. We propose a framework that composes pre-trained GFlowNets at inference time, enabling rapid adaptation without fine-tuning or retraining. Importantly, our framework is flexible, capable of handling diverse reward combinations ranging from linear scalarization to complex nonlinear operators, which are often handled separately in previous literature. We prove that our method exactly recovers the target distribution for linear scalarization, and quantify the approximation quality for nonlinear operators through a distortion factor. Experiments on a synthetic 2D grid and real-world molecule generation tasks demonstrate that our approach achieves performance comparable to baselines.

We study the problem of efficiently producing, in an online fashion, generative models of scalar, multiclass, and vector-valued outcomes that cannot be falsified on the basis of the observed data and a pre-specified collection of computational tests. Our contributions are twofold. First, we expand on connections between online high-dimensional multicalibration with respect to an RKHS and recent advances in expected variational inequality problems, enabling efficient algorithms for the former. We then apply this algorithmic machinery to the problem of outcome indistinguishability. Our procedure, Defensive Generation, is the first to efficiently produce online outcome indistinguishable generative models of non-Bernoulli outcomes that are unfalsifiable with respect to infinite classes of tests, including those that examine higher-order moments of the generated distributions. Furthermore, our method runs in near-linear time in the number of samples and achieves the optimal, vanishing T^{-1/2} rate for generation error.

Representation learning has been essential for graph machine learning tasks such as link prediction, community detection, and network visualization. Despite recent advances in achieving high performance on these downstream tasks, little progress has been made toward self-explainable models. Understanding the patterns behind predictions is equally important, motivating recent interest in explainable machine learning. In this paper, we present GraphHull, an explainable generative model that represents networks using two levels of convex hulls. At the global level, the vertices of a convex hull are treated as archetypes, each corresponding to a pure community in the network. At the local level, each community is refined by a prototypical hull whose vertices act as representative profiles, capturing community-specific variation. This two-level construction yields clear multi-scale explanations: a node's position relative to global archetypes and its local prototypes directly accounts for its edges. The geometry is well-behaved by design, while local hulls are kept disjoint by construction. To further encourage diversity and stability, we place principled priors, including determinantal point processes, and fit the model under MAP estimation with scalable subsampling. Experiments on real networks demonstrate the ability of GraphHull to recover multi-level community structure and to achieve competitive or superior performance in link prediction and community detection, while naturally providing interpretable predictions.

Monte Carlo rendering and modern generative models both transform uncertain states into structured images, yet they are usually studied as separate processes. We introduce Monte Carlo Transport Scheduling, a framework that treats progressive path tracing as a continuous sampling-driven transport process. Our key observation is that the renderer already produces physically valid states along this process: nested Monte Carlo estimates trace a refinement trajectory whose natural time coordinate follows from sampling variance. This view leads to a continuous training framework that learns from real render endpoints rather than synthetic interpolants, preserving the statistical structure of Monte Carlo estimation while enabling arbitrary-step neural refinement. We evaluate the framework on a controlled rendering benchmark designed to separate transport difficulty from scene context, and show that it yields stable render refinement, supports continuous stopping between rendering states, and transfers as a physical prior for frozen generative samplers. These results suggest a common continuous-time substrate for rendering and generation, where Monte Carlo sampling provides both the physical states and the supervision for learning image transport.

This paper provides a reinterpretation of the Drifting Model~\cite{deng2026generative} through a semigroup-consistent long-short flow-map factorization. We show that a global transport process can be decomposed into a long-horizon flow map followed by a short-time terminal flow map admitting a closed-form optimal velocity representation, and that taking the terminal interval length to zero recovers exactly the drifting field together with a conservative impulse term required for flow-map consistency. Based on this perspective, we propose a new likelihood learning formulation that aligns the long-short flow-map decomposition with density evolution under transport. We validate the framework through both theoretical analysis and empirical evaluations on benchmark tests, and further provide a theoretical interpretation of the feature-space optimization while highlighting several open problems for future study.

Flow-based generative methods offer a simple and effective framework for high-fidelity generation, yet pretrained flow models are rarely used in their vanilla conditional form: in image generation, samples without guidance often appear diffuse and lack fine-grained detail. Existing guidance techniques such as classifier-free guidance (CFG) improve fidelity but reduce sample diversity. We introduce Momentum Guidance (MG), a guidance method that improves sample quality by extrapolating the current velocity away from an exponential moving average of past velocities along the ODE trajectory, while preserving the standard one-evaluation-per-step cost. MG provides gains beyond CFG, improving the precision-recall Pareto frontier. Experiments demonstrate the effectiveness of MG across benchmarks. On ImageNet-256, MG improves FID by 36.54% without CFG and 25.42% with CFG on average across sampling settings, attaining an FID of 1.553 at 16 sampling steps. Evaluations on large flow-based models, including Stable Diffusion 3 and FLUX.1-dev, further confirm improvements across standard metrics.