Many statistical models involve parameter-dependent normalizing constants that are computationally intractable, creating substantial obstacles to standard Bayesian inference. Although existing likelihood-based algorithms can often circumvent these constants, their uncertainty quantification may be poorly calibrated under model misspecification. To address these challenges, we propose SME-BETEL, a semiparametric Bayesian framework that combines score matching estimating equations with Bayesian exponentially tilted empirical likelihood. The resulting posterior avoids evaluation of normalizing constants and does not require learning-rate calibration. We establish consistency and asymptotic normality of the score matching estimator, and prove a Bernstein-von Mises theorem for the SME-BETEL posterior. These results show that SME-BETEL credible sets are asymptotically calibrated to the sampling variability of the score matching estimator, yielding valid frequentist coverage under model misspecification. We further develop a new score matching criterion for mixed-domain data, extending SME-BETEL to models whose observations combine components from different sample spaces. This construction enables robust Bayesian inference for mixed-domain doubly-intractable models, including preferential sampling models with an intractable spatial normalizing integral. Simulation studies show that SME-BETEL remains competitive under correct specification and substantially improves uncertainty quantification under misspecification. An ozone-monitoring application further demonstrates the utility of the mixed-domain construction for spatial preferential modeling.
Near a smooth data manifold, one tangent space summarizes local geometry. At a branch point, the corresponding first-order object is instead a measure over tangent directions, whose normalized masses record the local share of each branch under the chosen data measure. We ask whether a score field at one noise level determines this weighted tangent geometry when the branch center and homogeneity degree $d$ are unknown. In this tangent-measure model, $d$ is the local measure dimension. Gaussian smoothing of a homogeneous tangent measure satisfies an Ornstein--Uhlenbeck eigenfunction equation. Its weak form turns score values---without score derivatives---into a linear system for the center and homogeneity degree, with an explicit rank condition and perturbation bound. After this calibration, the tangential score on one sphere is the spherical log-gradient of a scalar Gaussian--cone transform. Integration recovers that transform up to scale, and all its spherical-harmonic multipliers are positive. Thus one exact shell identifies the normalized angular measure in every ambient dimension $D\geq2$. For at most $K$ positive rays, moments through degree $2K-1$ constructively recover count, directions, and weights in arbitrary dimension. Any fixed observation scheme needs at least $KD-1$ scalar tangential components. In the plane, degree $K$ is both sufficient and necessary, and we give quantitative finite-query certificates. For finite planar $C^{1,β}$ branches with positive $C^{0,β}$ densities, we prove $O(σ^β)$ convergence from the finite-noise score to its tangent model. In controlled experiments, 50k-step training lowers validation normalized-score error across four geometries yet raises angular-moment error, separating ordinary score fit from geometry recovery.
Jakub Micorek, Mateusz Koziński, Horst Posseggercs.CV
Skeleton-based Video Anomaly Detection (VAD) offers a robust, privacy-preserving solution for identifying abnormal behaviors. To model the distribution of normal static and moving poses, recent methods train Energy-Based Models (EBMs) via Denoising Score Matching (DSM). However, directly injecting noise, required for training, into raw joint coordinates creates physically impossible poses, and this structural collapse severely worsens as the temporal window expands. To address this, we introduce STEP, a simple framework that utilizes Principal Component Analysis (PCA) to project pose sequences into a compact, whitened PC-space. Learning the data density within this well-behaved PC-space ensures that the injected noise translates into physically plausible variations, which allows the model to process longer video sequences without the performance collapse of raw coordinate baselines. Additionally, to mitigate inherent pose estimation inaccuracies arising from occlusions or motion blur, we integrate a sequence-level weighting mechanism based on the estimator's confidence scores. Operating at real-time computational efficiency, our simple and lightweight framework outperforms the previous skeleton-based state-of-the-art by 12.2% (90.1% AUROC) on the challenging UBnormal dataset and achieves highly competitive results by improving on the ShanghaiTech benchmark.
We study score-matching diffusion models with a convolutional architecture. We argue that the inductive bias of locality means that the machinery of effective field theory from physics can be usefully applied to describe the denoising dynamics. We apply this formalism first to a simple toy example which permits an analytical description, and thereafter to MNIST, and show that in both cases, the mutual information between two points grows in a manner predicted by a simple effective field theory of Brownian motion.
We formulate generative modeling as a path integral in which flow-based, diffusion-based, variational, and adversarial models arise as different evaluation principles for a single master action. Its Martin-Siggia-Rose-Janssen-de~Dominicis (MSRJD) form separates free from interacting probability flows and opens them to diagrammatic perturbation theory. The expansion yields a one-loop correction to deterministic samplers at no stochastic-sampling cost, which we validate on solvable and nonlinear drifts, where it reduces a 53 % tree-level error to 1.6 %. Imperfect learned scores enter as insertions and yield a response-weighted score-matching objective, and symmetry-equivariant drift design becomes an operator expansion with EFT power counting.
We study score learning for reflected diffusion on bounded domains. Reflection keeps trajectories feasible but does not ensure that the learned score satisfies the boundary behavior implied by the forward process. With implicit score matching, integration by parts leaves a boundary term, and we show that it depends on one scalar at each boundary point: the diffusion- weighted normal component of the score, or conormal trace. The no-flux condition fixes this value while leaving the re- maining boundary components unrestricted; under anisotropic diffusion it generally differs from the ordinary normal score component. On hyperrectangles, our parametrization enforces the required trace without additional trainable parameters or a stochastic boundary estimator and, under regularity assump- tions, can represent the true score, whereas fixing an incorrect value creates an error that more data cannot remove. We ex- tend the construction to simplices and polygonal domains and identify reflection masking: hard reflection can keep samples feasible even when the learned trace is wrong, so post-reflection metrics may hide the error. Experiments show the clearest separation with less frequent reflection, anisotropic diffusion, and mass near intersections of constraints; under full reflection, final sample placement improves inconsistently, illustrating how hard repair can mask boundary-score errors and decouple score accuracy from downstream generation quality.
The performance of generative diffusion models is determined by the choice of the reference diffusion process connecting the empirical and prior distributions. Conventional approaches typically trade off simulation-free training against finite-time generation. We propose a framework for designing the reference process that achieves both simultaneously. The key idea is to prescribe tractable time-dependent conditional distributions and then construct the reference process realizing them as its marginals. This framework reveals that score matching is not fundamental to diffusion-model training but instead emerges naturally through reversal of the reference process. We further show that conditional flow matching arises as the small-noise limit of the proposed framework.
Score-based diffusion models are typically formulated using continuous-time stochastic differential equations and measure-theoretic stochastic calculus. In this paper, we develop a hyperfinite formulation of score-based generative modeling within the framework of Nonstandard Analysis. Starting from an internal diffusion process on a hyperfinite grid, we derive the associated infinitesimal generator and establish its correspondence with the classical Fokker--Planck equation. We then obtain a hyperfinite backward-mean identity that yields the reverse-time drift and provides a constructive derivation of the reverse-time SDE. Building on these results, we show that minimization of an internal score-matching objective recovers the score function required by the reverse-time dynamics, thereby connecting score estimation with generative sampling directly at the hyperfinite level. Under suitable assumptions, we further derive a hyperfinite Girsanov formula and establish a relationship between likelihood optimization and Fisher-divergence objectives. Finally, we analyze the second-order consistency of the hyperfinite dynamics and show that the leading correction term depends explicitly on the fourth moment of the increment distribution, with the Gaussian value $κ=3$ eliminating the leading dispersion contribution. Taken together, these results provide a unified hyperfinite framework for diffusion-based generative modeling--while laying foundations for further extensions--that links discrete grid dynamics, reverse-time diffusion, score matching, and likelihood-based formulations within a common nonstandard setting.
Score matching controls average error under the forward marginals, but a discretized reverse-time sampler evaluates the learned score along its own trajectory. We show that small forward-marginal error does not guarantee numerical stability. We construct a single smooth score field with arbitrarily small forward-marginal $L^2$ error. The learned reverse-time process is nonexplosive, has moments of every order, and can be arbitrarily close to the exact reverse-time process in path-space total variation. Yet its Euler--Maruyama discretizations converge in probability while every positive moment diverges. Thus weak convergence can hold even though every Wasserstein distance $W_p$, $p\ge1$, diverges. The same failure can occur within one fixed finite neural architecture. We construct a family of bounded, globally Lipschitz denoisers for which both the forward-marginal error and the path-space total variation distance tend to zero, while their Euler--Maruyama endpoints diverge in every $W_p$. For compactly supported data, we also give a simple positive result. Projecting the learned denoiser onto a known bounded closed convex set containing the support preserves pointwise accuracy, gives grid-uniform moment bounds, and yields Wasserstein convergence under mild local regularity. Experiments with a small fixed DiT-style network show large growth along rare numerical trajectories and its suppression by denoiser projection, while overall trajectory errors remain small.
Benjamin Dupuis, Tyler Farghly, Maxime Haddouche +2stat.ML cs.LG
Diffusion models (DMs) are a state-of-the-art generative method to approximately sample from an unknown distribution. Their training and evaluation primarily rely on an Evidence Lower Bound (ELBO), which relates the Kullback-Leibler (KL) divergence of model samples to the score matching loss along the path, which serves as a tractable surrogate. The difference between sample quality and the score matching loss produced by this bound leads to the \emph{score matching gap}, which is known to be tight in the worst-case but not descriptive of sample quality in general. In this work, we provide a theoretical analysis of this gap, developing tighter bounds for three metrics: KL divergence, reverse KL divergence, and Wasserstein distance, effectively exploiting the regularity of the class of score estimators. Our results suggest that the quality of the score approximation has more impact on closing the score matching gap for low noise scales. To obtain these bounds, our key technical insight is to exploit the contraction properties of the backward processes. In particular, we rely on entropy flows, logarithmic Sobolev inequalities and reflection couplings, rigorously linking the ergodicity of the Langevin diffusion to the score matching gap problem.
Tyler Farghly, Benjamin Dupuis, Alain Durmus +1stat.ML cs.LG
Benign overfitting and double descent have come to shape our understanding of generalization in deep learning, establishing that overfitting is not only compatible with good generalization but can actively benefit it. Diffusion models share much of the machinery of standard deep learning, so it is natural to assume that they also exhibit these properties. In this work, we show that this assumption is largely incorrect. We first establish fundamental impossibility results showing that, unless the sample size grows exponentially with the data dimension, overfitting and good generalization cannot occur simultaneously. Consequently, the population loss follows a classical U-shaped curve in model complexity rather than exhibiting double descent. Analyzing a simplified setting, we identify a key difference between regression and score matching: regression benefits from an alignment between the target and the empirical covariance; score matching admits no such alignment, leaving overfitting irreparably harmful. We further identify implicit regularization stemming from time-smoothness of the score and early stopping during training as mechanisms that prevent such overfitting and verify our findings with high-dimensional image generation experiments. Our results reveal that generalization in diffusion models is governed by mechanisms distinct from those of traditional regression, motivating the development of new theory.
The use of ordinary and stochastic differential equations has led to substantial progress in generative machine learning with applications to, for example, image, video and biomolecule generation. This paper provides a self-contained and informal introduction to the differential equations, the probabilistic framework for using them in generative modeling and the Fokker--Planck equation that governs the temporal evolution of the marginal distribution of the stochastic variables of the differential equations. The variational lower bound on the log-likelihood (the evidence lower bound, ELBO) is derived and used as a general starting point for a discussion of diffusion models, score matching, and flow matching. All of these approaches may be viewed as specific parameterizations of the most general variational approach. A one-dimensional density modeling problem is used as a simple example to compare different parameterizations.
Characterizing non-Gaussian posterior distributions in partially observed high-dimensional nonlinear systems remains a fundamental challenge in data assimilation. Ensemble Kalman filters rely on Gaussian approximations that can be inaccurate for strongly non-Gaussian posteriors, whereas particle filters suffer from severe scalability limitations. Recent score-based generative approaches improve posterior characterization but typically require supervised training with ground-truth posterior samples, which are unavailable in most practical applications. We introduce $Ω$ (Operator-based Mixture Ensemble for Generative Assimilation), a scalable framework that integrates conditional Gaussian surrogate modeling, unsupervised score learning, and generative sampling. The conditional Gaussian surrogate provides a nonlinear non-Gaussian baseline approximation while admitting closed-form conditional posterior distributions for the unresolved variables. First, $Ω$ exploits these closed-form conditional distributions to analytically recover the high-dimensional unobserved component, reducing computational cost and mitigating the curse of dimensionality. Second, $Ω$ learns only the residual discrepancy beyond an analytical baseline through denoising score matching using ensemble trajectories alone, eliminating the need for ground-truth posterior samples and substantially reducing the learning burden. Third, $Ω$ reconstructs the full non-Gaussian posterior distribution of both observed and unobserved variables via a Gaussian mixture representation, capturing multimodal, skewed, and heavy-tailed statistics. Finally, $Ω$ employs annealed Langevin sampling to iteratively refine ensemble members from the baseline toward the target posterior. $Ω$ is validated on several turbulent models with intermittency and extreme events, consistently improving posterior accuracy.
The score matching problem is a central training objective in modern generative modeling, diffusion models, fitting unnormalized statistical models, and inverse problems. A standard approach is to minimize the forward Fisher divergence, where the expectation is taken with respect to the teacher distribution. However, recent results show that even in simple Gaussian mixture model settings, this objective can lead to undesirable and initialization-dependent convergence behavior. In this paper, we study an alternative objective: the reverse Fisher divergence, where the expectation is taken with respect to the student distribution. We analyze gradient descent (GD) for fitting Gaussian mixture models and show that this change in the objective leads to significantly better optimization properties. First, when the teacher distribution is a single Gaussian and the student is a Gaussian mixture model with fixed weights and identity covariances, we prove the global convergence of GD from arbitrary initializations. Second, we extend the analysis to the case where the teacher is also a Gaussian mixture model and prove global convergence guarantees under a global random initialization scheme and a $\widetildeΩ(1)$-separation assumption on the target means. In particular, with high probability, each student component converges near its closest teacher component, and we provide conditions under which the student distribution converges in total variation distance. Our proofs rely on a new Lyapunov-based analysis of the gradient descent dynamics, showing that the reverse Fisher divergence has a much more favorable optimization landscape than the forward Fisher divergence.
Naïl B. Khelifa, Richard E. Turner, Ramji Venkataramananstat.ML cs.LG
Score-based diffusion models are typically trained by minimizing the $L^2$ score matching error, and standard theoretical analyses rely on this quantity to bound the sampling discrepancy between the learned and target distributions. We show the $L^2$ score error is not the right intrinsic measure of marginal distributional quality: a learned diffusion model can incur arbitrarily large $L^2$ score error while perfectly matching the target distribution. By decomposing score errors into a gradient and a solenoidal component (a Helmholtz-Hodge decomposition), we identify the geometric reason behind this: only the gradient component enters the marginal Fokker-Planck dynamics, while the solenoidal component is structurally invisible. We make this precise in three results. First, building on the corrected geometry, we prove an impossibility result: no monotone function of the $L^2$ score error can uniformly lower bound any divergence between the learned and target distributions. Second, we derive an upper bound on the Kullback-Leibler divergence that depends only on the observable gradient component of the error, tightening the standard Girsanov bound for generic score networks, and identifying its looseness as the cost of operating on path-space rather than marginal-space dynamics. Third, we give a tractable estimator of the gradient component via a dual Sobolev identity, which is shown to empirically correlate substantially better with sample quality than the full $L^2$ error.
Generating realistic synthetic sequential data is critical in real-world applications across operations research, finance, healthcare, energy systems, and scientific computing, where time-indexed observations are used for prediction, simulation, risk assessment, and data-driven decision-making. While diffusion models have achieved remarkable success in generating static data, their direct extensions to sequential settings often fail to capture temporal dependence and information structure. Designing diffusion models that can simulate sequential data in an adapted manner, and hence without anticipation of future information, therefore remains an open challenge. In this work, we propose a sequential forward-backward diffusion framework for adapted time series generation. Our approach progressively injects and removes noise along the sequence, conditioning on the previously generated history to ensure adaptiveness. A novel score-matching objective is introduced for efficient parallel training. We derive rigorous statistical guarantees under a generic framework, then establish score approximation, score estimation, and distribution estimation results with ReLU networks serving as a concrete instance. Empirically, we validate our method on synthetic data, including ARMA models and Gaussian processes, and demonstrate its effectiveness in constructing mean-variance optimal portfolios.
Parameter estimation in stochastic differential equations is a classical statistical problem of much importance in many scientific fields. Recent work of Tapia Costa et al. (2026) introduced a novel technique for estimating the drift when the diffusion parameter is known, using discrete samples from multiple trajectories. Their method treats drift estimation as a denoising problem, and leverages tools from (conditional) score-matching diffusion models. Although their experiments showed promising results across different drift classes, the question of theoretical guarantees for their estimator was left unanswered. In this note, we address this gap by exploiting techniques from diffusion model theory. More concretely, we derive an explicit risk bound for the time-averaged mean-squared error of said drift estimator. Our bound decomposes the risk into the (i) Euler-Maruyama discretization, (ii) score/denoiser approximation, (iii) noise initialization, and (iv) sampling variance, revealing the trade-offs between the different hyperparameters and sources of error in the estimator.
This tutorial develops diffusion models from the viewpoint of differential equations. We begin with the conditional Gaussian forward process and show that this path admits both an ordinary differential equation (ODE) representation and a stochastic differential equation (SDE) representation. Averaging the conditional process over the data distribution then yields marginalized forward ODE and SDE formulations that transport the data distribution $p_0=p_{\mathrm{data}}$ to a Gaussian prior $p_1=\mathcal{N}(0,I)$. We next derive the corresponding reverse-time dynamics, namely the reverse SDE and the reverse probability-flow ODE, both of which are governed by the marginal score $\grad\log p_t(x)$. This leads to a training objective for score estimation and shows that the standard noise-prediction objective is equivalent to score matching up to an additive constant independent of the model parameters. We then discuss sampling methods for the learned reverse dynamics, including DPM-Solver, as well as guided sampling through classifier guidance and classifier-free guidance. Finally, we compare DDPM and DDIM with the reverse SDE/ODE framework and show that they share the same training objective, while DDPM sampling corresponds to discrete reverse-SDE sampling and DDIM sampling corresponds to reverse-ODE sampling.
Gabe Guo, Thanawat Sornwanee, Lutong Hao +3cs.LG cs.AI
Generating continuous-time, continuous-space stochastic processes (e.g., videos, weather forecasts) conditioned on partial observations (e.g., first and last frames) is a fundamental challenge. Existing approaches, (e.g., diffusion models), suffer from key limitations: (1) noise-to-data evolution fails to capture structural similarity between states close in physical time and has unstable integration in low-step regimes; (2) random noise injected is insensitive to the physical process's time elapsed, resulting in incorrect dynamics; (3) they overlook conditioning on arbitrary subsets of states (e.g., irregularly sampled timesteps, future observations). We propose ABC: Any-Subset Autoregressive Models via Non-Markovian Diffusion Bridges in Continuous Time and Space. Crucially, we model the process with one continual SDE whose time variable and intermediate states track the real time and process states. This has provable advantages: (1) the starting point for generating future states is the already-close previous state, rather than uninformative noise; (2) random noise injection scales with physical time elapsed, encouraging physically plausible dynamics with similar time-adjacent states. We derive SDE dynamics via changes-of-measure on path space, yielding another advantage: (3) path-dependent conditioning on arbitrary subsets of the state history and/or future. To learn these dynamics, we derive a path- and time-dependent extension of denoising score matching. Our experiments show ABC's superiority to competing methods on multiple domains, including video generation and weather forecasting.
Lei Wangquant-ph cs.LG physics.chem-ph physics.comp-ph
We solve the time-dependent Schrödinger equation by learning the score function, the gradient of the log-probability density, on Bohmian trajectories. In Bohm's formulation of quantum mechanics, particles follow deterministic paths under the classical potential supplemented by a quantum potential depending on the score function of the evolving density. These non-crossing Bohmian trajectories form a continuous normalizing flow governed by the score. We parametrize the score with a neural network and minimize a self-consistent Fisher divergence between the network and the score of the resulting density. We prove that the zero-loss minimizer of this self-consistent objective recovers Schrödinger dynamics for nodeless wave functions, a condition naturally met in quantum vibrations of atoms. We demonstrate the approach on wavepacket splitting in a double-well potential and anharmonic vibrations of a Morse chain. By recasting real-time quantum dynamics as a self-consistent score-driven normalizing flow, this framework opens the time-dependent Schrödinger equation to the rapidly advancing toolkit of modern generative modeling.