Score-Based Generative Modeling

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6 papers in the last four weeks, with none the four weeks before. 0.1% of all new papers.

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Latest papers 71

May 29, 2026cs.LG

Riemannian Diffusion Models on General Manifolds via Physics-Informed Neural Networks

Riemannian diffusion models generalize score-based generative modeling to manifold-supported data via stochastic diffusion equations on the manifold. However, training requires sampling from and differentiating the manifold heat kernel, which is rarely available in closed form beyond a few highly symmetric manifolds. We propose a general approach that approximates the heat kernel by directly solving the manifold heat equation with a physics-informed neural network (PINN). Given an explicit manifold specification, we choose a coordinate system, derive the corresponding heat (Fokker--Planck) equation and a short-time asymptotic approximation, and then train a PINN to learn the log heat kernel. The resulting surrogate enables both forward noising (heat-kernel sampling) and conditional-score evaluation for denoising score matching. We demonstrate the method on diverse manifolds including

S^2

,

SO(3)

,

\mathrm{SPD}(n)

, and permutation-quotiented point clouds.

Gyeonghoon Ko, Juho Lee

1Korea Advanced Institute of Science and Technology, Korea.

Physics-Informed Diffusion Models Physics-Informed Generative Modeling Diffusion Model Sampling Score-Based Generative Modeling Diffusion Models

May 28, 2026math-ph

The Score Hamiltonian: Mapping Diffusion Models to Adiabatic Transport

We exhibit an exact correspondence between sampling with score-based diffusion models and adiabatic transport of ground states for a family of Schrödinger operators we call Score Hamiltonians, built from the learned score's quantum potential. We obtain novel density reconstruction bounds and principled annealing schedules via adiabatic theorems for Fokker-Planck equations with time-varying potentials. We find the fundamental limit of sampling is set by the ratio of squared score-matching error to Score Hamiltonian spectral gap - the inverse Poincaré constant of the data density.

Peter Halmos, Boris Hanin

Computer Science Department, Princeton University. · ORFE Department, Princeton University.

Diffusion Sampling Score-Based Generative Modeling

May 26, 2026cs.LG

GenSBI: Generative Methods for Simulation-Based Inference in JAX

Flow and diffusion generative models have established themselves as widely adopted density estimators for simulation-based inference (SBI), extending naturally from neural posterior estimation to likelihood and joint density estimation. Their principled optimization objectives and freedom from architectural constraints have driven rapid adoption across the natural sciences. Yet the most widely used SBI libraries remain PyTorch-based, leaving researchers who develop their forward models and analysis pipelines in JAX without a native option. We present GenSBI, an open-source library that implements flow matching, score matching, and denoising diffusion entirely in JAX. The library offers three transformer-based architectures - SimFormer, Flux1, and a novel Flux1Joint that extends gate-modulated transformer blocks to joint density estimation - all interchangeable through a unified interface that decouples generative method, neural backbone, and inference mode. GenSBI provides an end-to-end workflow from training through posterior calibration (SBC, TARP, LC2ST) and supports custom architectures with domain-specific embedding networks. We validate the framework on standard SBI benchmarks, achieving near-ideal mean C2ST scores (0.50-0.56, where 0.50 is ideal) on SBIBM tasks with minimal per-task tuning and well-calibrated posterior coverage across all tested configurations. The code is publicly available at https://github.com/aurelio-amerio/GenSBI.

Aurelio Amerio

Instituto de Física Corpuscular (IFIC) Universitat de València & CSIC València, Spain

Flow Matching Simulation-Based Inference Neural Posterior Estimation Flow-Based Generative Modeling Score-Based Generative Modeling Diffusion Models

May 25, 2026stat.ML

Rao-Blackwellized Score Matching on Manifolds

We study denoising score matching (DSM) when the latent distribution is supported on a smooth embedded manifold

M \subset \mathbb{R}^D

. Under ambient Gaussian corruption, the tangent denoising target contains a singular normal-fiber noise channel whose variance diverges as

d/σ^2

as

σ\to 0^+

. We show that conditioning on the nearest-point projection

π(X)

canonically removes this singularity: the resulting conditional expectation is the unique

L^2

-optimal Rao-Blackwellized predictor of the tangent DSM target among all estimators depending only on the projected observation

π(X)

. We then compute the small-noise expansion of this canonical target and show that it equals the intrinsic Riemannian score up to an explicit order-

σ^2

correction that decomposes into an intrinsic Tweedie term and an extrinsic curvature term involving the Weingarten and Ricci operators. In the flat case, the construction reduces exactly to ordinary lower-dimensional Gaussian DSM, while on

S^d

the extrinsic correction simplifies to the scalar factor

(1-d/2)\nabla_M \log q

; this extrinsic

σ^2

correction cancels identically on

S^2

, though the intrinsic Tweedie term remains.

Divit Rawal

Department of Statistics, University of California, Berkeley, Berkeley CA, U.S.A..

Score Matching Score-Based Generative Modeling

May 21, 2026stat.ML

Diffusion-based Denoising Beats Vanilla Score Matching in Parameter Estimation: A Theoretical Explanation

Score matching is an alternative to maximum likelihood estimation when the normalizing constant is unknown or too costly to evaluate. However, vanilla score matching has shown to be inefficient relative to maximum likelihood estimation for multimodal distributions with well-separated modes, which are commonly encountered in practical applications. We compare a novel diffusion-based denoising score matching estimator (DDSME) to the vanilla score matching estimator (SME) in this scenario. In particular, we prove statistical guarantees for both estimators, showing that the error bound for the vanilla SME worsens when the separation between the modes increases, which can be avoided in case of the DDSME with suitable hyperparameter tuning. This provides a novel theoretical explanation for the superior behavior of diffusion-based score matching over the vanilla version.

Benedikt Lütke Schwienhorst, Nadja Klein, Johannes Lederer

Department of Mathematics University of Hamburg Hamburg, Germany · Scientific Computing Center Karlsruhe Institute of Technology2026 Karlsruhe, Germany · Department of MathematicsMay University of Hamburg Hamburg, Germany21

Score Matching Parameter Estimation Score-Based Generative Modeling Diffusion Models

May 21, 2026cs.LG

A Tutorial on Diffusion Theory: From Differential Equations to Diffusion Models

Diffusion models have emerged as a dominant framework for generative modeling, but their mathematical foundations are often presented separately through diffusion probabilistic models, score-based modeling, stochastic differential equations, and numerical sampling methods. We write this tutorial to provide a unified and self-contained account of these viewpoints from the perspective of differential equations. Starting from a conditional Gaussian noising process, we derive ordinary differential equation (ODE) and stochastic differential equation (SDE) representations, pass to the corresponding marginal forward dynamics, and then obtain the reverse-time SDE and probability-flow ODE that make generation possible. We show that the central unknown quantity in reverse sampling is the marginal score, explain how score matching becomes the standard denoising objective under a noise-prediction parameterization, and discuss practical reverse-time sampling and guidance. We further place DDPM, DDIM, flow matching, and score-based SDEs in a common framework, and conclude with diffusion language models in continuous embedding space together with a brief discussion of discrete masked-token diffusion. The tutorial is intended as a bridge between the analytical foundations of diffusion processes and the modern generative algorithms built upon them.

Jiayi Fu, Yuxia Wang

INSAIT, Sofia University “St. Kliment Ohridski”, Bulgaria

Diffusion Model Sampling Score-Based Generative Modeling Diffusion Models

May 21, 2026cs.LG

Physics-Informed Generative Solver: Bridging Data-Driven Priors and Conservation Laws for Stable Spatiotemporal Field Reconstruction

Reconstructing continuous physical fields from sparse measurements is a central inverse problem, but data-driven generative models can produce states that violate governing dynamics. We introduce a physics-informed generative solver that separates stable prior learning from inference-time enforcement of conservation laws. Martingale-Regularized Score Matching regularizes score pretraining with a Score Fokker-Planck constraint, yielding a dynamically stable prior. Physics-Informed Implicit Score Sampling then guides denoising trajectories by gradients of physical residuals, projecting samples toward admissible manifolds without retraining. In acoustics, the method co-generates pressure and particle velocity from sparse sensors, enabling dense virtual arrays that suppress spatial aliasing. The same framework generalizes to real-world ERA5 meteorological fields under extreme sparsity. Together, this work establishes a rigorous and generalizable paradigm for solving high-dimensional inverse problems, bridging the gap between generative artificial intelligence and first-principles science.

Ziyuan Zhu, Keyu Hu, Zhifei Chen +10

School of Advanced Manufacturing Engineering, Nanjing University, Suzhou, 215163, China. · National Laboratory of Solid State Microstructures, Nanjing University, Nanjing, 210093, China. · Suzhou Acoustics Industry Technology Research Institute Co., Ltd., Suzhou, 215513, China. +8

Physics-Informed Generative Modeling Physics-Informed ML Score-Based Generative Modeling PDE Inverse Problems Inference-Time Guidance

May 19, 2026stat.ML

Tweedie's Formulae and Diffusion Generative Models Beyond Gaussian

Diffusion models have achieved remarkable success in generating samples from unknown data distributions. Most popular stochastic differential equation-based diffusion models perturb the target distribution by adding Gaussian noise, transforming it into a simple prior, and then use denoising score matching, a consequence of Tweedie's formula, to learn the score function and generate clean samples from noise. However, non-Gaussian diffusion models with state-dependent diffusion coefficient have been largely underexplored, as have the corresponding Tweedie's formulae. In this work, we extend Tweedie's formula to important non-Gaussian processes, including geometric Brownian motion (GBM), squared Bessel (BESQ) processes, and Cox-Ingersoll-Ross (CIR) processes, thereby yielding the corresponding denoising score-matching objectives. We then apply the derived formulae to image and financial time series generation using GBM- and CIR-based diffusion models, and to empirical Bayes estimation under the BESQ setting. The reported experimental results demonstrate the potential of non-Gaussian models.

Wenpin Tang, Nizar Touzi, Zikun Zhang +1

Stochastic Differential Equations Time Series Generation Score-Based Generative Modeling Diffusion Models

May 18, 2026stat.ML

Reducing Diffusion Model Memorization with Higher Order Langevin Dynamics

Diffusion/score-based models have emerged as powerful generative models, capable of generating high-quality samples that mimic the training data distribution. However, it has been observed that they are prone to reproducing training samples-known as "memorization"-potentially violating copyright and privacy. In this paper, we study the effect of Higher-Order Langevin Dynamics (HOLD) on this phenomenon. HOLD diffusion processes introduce auxiliary variables; if the data variable is interpreted as "position," then the auxiliary variables can be interpreted as "velocity" and "acceleration," depending on the chosen order of the model. They were originally proposed based on the intuition that they regularize the trajectories of the data variable by implicitly imposing additional dynamical constraints. Our work provides, to our knowledge, the first theoretical characterization of the regularization effect of HOLD. Specifically, we show that in HOLD, the dynamics of the data variable are governed by a low-pass-filtered version of the learned score function, with smoothness increasing with the order of HOLD. We then analyze the optimal empirical score and the possibility of distribution collapse. Together, our results explain the mitigation of memorization as the model order increases. Finally, we present an empirical study on real-world data that supports our theory and highlights this distinct advantage of HOLD over standard diffusion in practice.

Benjamin Sterling, Mónica F. Bugallo, Tom Tirer

Department of Applied Math & Statistics Stony Brook University Stony Brook, NY 11790 · Department of Electrical and Computer Engineering Stony Brook University Stony Brook, NY 11790 · Faculty of Engineering Bar-Ilan University Ramat-Gan 5290002, Israel

Langevin Dynamics Memorization in Generative Models Score-Based Generative Modeling Diffusion Models

May 16, 2026cs.LG

Provably Learning Diffusion Models under the Manifold Hypothesis: Collapse and Refine

Diffusion models generate high-dimensional data with remarkable quality, yet how their training efficiently learns the score function, bypassing the curse of dimensionality when data is supported on low-dimensional manifolds, remains theoretically unexplained. We identify a collapse-and-refine mechanism driven by the geometry of the score function itself: at small noise scales, the diverging singularity of the score drives a rapid dimensional collapse of the induced denoising map onto the data manifold projection; at moderate noise scales, training refines the intrinsic density on the learned manifold. We instantiate this principle as Score-induced Latent Diffusion (SiLD), a two-stage framework in which both manifold learning and density estimation emerge from a single denoising score matching objective, replacing the heuristic KL regularization of VAE-based latent diffusion models. We prove that the resulting sample complexity depends on the intrinsic dimension rather than the ambient dimension. Experiments on Stacked MNIST, CelebA variants, and molecular generation benchmarks show that SiLD matches or outperforms VAE-based LDMs in generation quality and consistently improves reconstruction, validating our theoretical predictions.

Wei Huang, Andi Han, Mingyuan Bai +4

RIKEN AIP & The Institute of Statistical Mathematics · University of Sydney · Agency for Science, Technology and Research & The Institute of Statistical Mathematics +4

Latent Diffusion Models Manifold Learning Score-Based Generative Modeling Diffusion Models

May 16, 2026cs.CV

Towards Generalized Image Manipulation Localization via Score-based Model

With the rapid evolution of synthetic media, Image Manipulation Localization (IML) has emerged as a critical component in multimedia forensics for ensuring the integrity of digital content. However, generalization remains a core challenge, as existing discriminative methods typically learn a fixed decision boundary that tends to overfit to specific training artifacts and fails to adapt to unseen manipulation types. To address this, we propose DiffIML, a novel framework that introduces score-based generative modeling to IML. Diverging from the direct estimation of hard boundaries, DiffIML approximates the score function, the gradient of the log-likelihood, to capture the intrinsic geometric topology of mask distributions. This paradigm leverages structural priors to iteratively recover coherent masks from noise, thereby circumventing the brittleness associated with discriminative models. Under this formulation, diffusion models serve as an effective numerical solver for the learned score function.To ensure practicality, we respectively resolve the efficiency and stability bottlenecks of standard diffusion by: (1) utilizing a Lightweight Mask-Specific VAE for fast latent-space process and a decoupled architecture with a lightweight denoising UNet, (2) edge supervision and error prior to mitigate error accumulation during sampling. Extensive experiments of two distinct protocols on eight non-generative and three generative benchmarks demonstrate that DiffIML consistently outperforms state-of-the-art methods, yielding remarkable generalization improvements on diverse unseen datasets. The code is publicly available at https://github.com/scu-zjz/DiffIML.

Yunfei Wang, Bo Du, Zhe Yang +4

Sichuan University · Chengdu, China

Image Forgery Detection Image Forgery Localization AI-Generated Image Detection Score-Based Generative Modeling Diffusion Models

May 15, 2026cs.LG

Intrinsic Wasserstein Rates for Score-Based Generative Models on Smooth Manifolds

Score-based generative models are trained in high-dimensional ambient spaces, yet many data distributions are supported on low-dimensional nonlinear structures. We prove that, for compact

d

-dimensional smooth manifolds

\mathcal{M} \subset [0,1]^D

with

d > 2

and

β

-Hölder densities strictly positive on

\mathcal{M}

, a variance-preserving SGM estimator attains the intrinsic Wasserstein--1 sample exponent

\tilde{\mathcal{O}}(D^{\mathcal{O}_β(d)}n^{-(β+1)/(d+2β)})

, up to logarithmic factors and explicit geometry and density factors. The full nonasymptotic bound explicitly isolates the finite-order geometry envelope, Hölder radius, density lower bound, ambient dependence, and finite-order correction terms. The analysis separates score approximation into a large-noise tangent-cell regime and a small-noise projection-centered, de-Gaussianized Laplace regime. The key technical ingredient is a ReLU implementation of nearest-projection coordinates via finite intrinsic anchors and Gauss--Newton iterations, rather than approximating the manifold projection as a black-box high-dimensional smooth map. Consequently, for families with polynomially controlled geometry and density lower bounds, the constructed score-network parameters have polynomial ambient dependence.

Guoji Fu, Taiji Suzuki, Wee Sun Lee +1

National University of Singapore · The University of Tokyo · RIKEN Center for Advanced Intelligence Project +2

Intrinsic Dimensionality Generative Modeling Score-Based Generative Modeling Wasserstein Distance

May 13, 2026math.OC

Proximal-Based Generative Modeling for Bayesian Inverse Problems

Score-based diffusion models demonstrate superior performance in generative tasks but encounter fundamental bottlenecks in inverse problems due to the analytical intractability of the time-dependent likelihood score. To bridge this gap, we propose a novel proximal-based generative modeling (PGM) framework that rigorously circumvents explicit likelihood evaluation. Our framework is built upon a theoretical equivalence between Gaussian convolution in diffusion processes and Moreau-Yosida regularization in nonsmooth optimization. This enables a new sampling mechanism driven by the proposed Moreau score, which admits a closed-form expression via proximal operators. Moreover, we introduce Moreau score matching to learn the proximal operators that rely solely on samples drawn from the prior distribution. Theoretically, PGM eliminates the early-stopping bias inherent in the score-based diffusion model and achieves non-asymptotic convergence. Experiments demonstrate that PGM significantly surpasses state-of-the-art methods in reconstruction quality and sampling time.

Boyang Zhang, Zhiguo Wang, Ya-Feng Liu

School of Advanced Interdisciplinary Sciences, University of Chinese Academy of Sciences, Beijing, China · Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China · School of Mathematics, Sichuan University, Chengdu, China +2

Score Matching Bayesian Inverse Problems Generative Modeling Diffusion-Based Inverse Problems Score-Based Generative Modeling

May 11, 2026cs.LG

Free Energy Manifold: Score-Based Inference for Hybrid Bayesian Networks

We introduce the Free Energy Manifold (FEM), a score-trained conditional energy model specialized for inference in hybrid Bayesian networks with discrete and continuous variables. FEM represents each conditional factor as an energy landscape over learned discrete-parent embeddings and continuous observations, enabling posterior evaluation, generative sampling, and compositional inference across multiple continuous leaves by energy addition under conditional independence. A central finding is the mode-bridge artifact: standard conditional energy models can create low-energy ridges between separated modes of the same class, producing overconfident posteriors at off-data interior points. We analyze this failure and propose valley regularization, an off-data calibration term that restores near-uniform posteriors in such regions while preserving in-data fit. Across synthetic multimodal hybrid-BN benchmarks, FEM substantially reduces KL divergence relative to classical baselines and a vanilla conditional EBM, including large gains at mode-bridge midpoint queries and in multi-leaf evidence composition. We also evaluate high-cardinality discrete-parent settings and a UCI Breast Cancer sanity check, showing that FEM is most useful when multimodal or compositional Bayesian-network inference is required, while discriminative classifiers remain preferable for closed-world classification tasks.

Cheol Young Park, Shou Matsumoto

ATOS Co., Ltd., 273, Digital-ro, Guro-gu, Seoul, 08381, Republic of Korea · C5I Center, George Mason University, Fairfax, VA 22030, USA

Bayesian Networks Energy-Based Models Conditional Distribution Estimation Score-Based Generative Modeling Uncertainty Calibration

May 10, 2026cs.LG

Learning Unified Representations of Normalcy for Time Series Anomaly Detection

The core challenge in unsupervised anomaly detection is identifying abnormal patterns without prior knowledge of their characteristics. While existing methods have addressed aspects of this problem, they often struggle to learn a robust representation of the normal data distribution that is distinct from anomalous patterns. In this paper, we present a novel framework, Unified Unsupervised Anomaly Detection (

\text{U}^2\text{AD}

), that comprehensively addresses anomaly detection in multivariate time series. Our approach learns the underlying data distribution of normal samples by utilizing score-based generative modeling. We introduce a novel time-dependent score network and a unified training objective that together delineate the manifold of normal data while considering both local and global temporal contexts. Reconstruction is then performed via a deterministic sampling process using an ordinary differential equation solver. Our extensive experimental evaluations demonstrate that

\text{U}^2\text{AD}

not only outperforms current state-of-the-art methods in detection accuracy but also identifies anomalies at significantly earlier stages of their occurrence.

Prithul Sarker, Sushmita Sarker, Nicholas G. Murray +1

University of Nevada, Reno

Reconstruction-Based Anomaly Detection Unsupervised Anomaly Detection Score-Based Generative Modeling Time-Series Anomaly Detection

May 10, 2026stat.ML

Metropolis-Adjusted Diffusion Models

Sampling from score-based diffusion models incurs bias due to both time discretisation and the approximation of the score function. A common strategy for reducing this bias is to apply corrector steps based on the unadjusted Langevin algorithm (ULA) at each noise level within a predictor-corrector framework. However, ULA is itself a biased sampler, as it discretises a continuous diffusion process. In this work, we consider adjusted Langevin correctors that employ Metropolis--Hastings (MH) or Barker's accept-reject steps to correct for this bias. Since the target density ratio typically required by MH-based algorithms is unavailable, we propose methods that instead utilise the score function to compute the correct acceptance probability. We introduce the first exact method for adjusting Langevin corrections in diffusion models, based on a two-coin Bernoulli factory algorithm. We also propose an efficient approximation based on Simpson's rule that achieves accuracy of order

5/2

in the step size at near-zero marginal cost. We demonstrate that these procedures improve sample quality on both synthetic and image datasets, yielding consistent gains in Fréchet Inception Distance (FID) on the latter.

Kevin H. Lam, Tyler Farghly, Christopher Williams +3

Department of Statistics, University of Oxford · Inria, CNRS, École Normale Supérieure, PSL Research University · Department of Mathematical Sciences, University of Copenhagen

Markov Chain Monte Carlo Diffusion Sampling Score-Based Generative Modeling

May 8, 2026cs.LG

Slowly Annealed Langevin Dynamics: Theory and Applications to Training-Free Guided Generation

We study Slowly Annealed Langevin Dynamics (SALD), a sampler for tracking a path of moving target distributions and approximating the terminal target through time slowdown. We establish non-asymptotic convergence guarantees via a KL differential inequality, showing that slowdown improves tracking through contraction of intermediate targets and the complexity of the path. Motivated by training-free guided generation with pretrained score-based generative models, we further introduce Velocity-Aware SALD (VA-SALD), which explicitly incorporates the underlying marginal distributions of the pretrained model and uses slowdown to correct the additional deviation induced by guidance. This yields a principled framework for training-free guided generation for diffusion-based and related generative model families, together with convergence guarantees that clarify the roles of intermediate functional inequalities and guidance bias. Code is available at https://github.com/anitan0925/sald.

Atsushi Nitanda, Dake Bu, Yueming Lyu +1

Agency for Science, Technology and Research (A⋆STAR) · Nanyang Technological University · City University of Hong Kong

Diffusion Model Guidance Langevin Dynamics Score-Based Generative Modeling

May 8, 2026cs.LG

Generative Modeling with Flux Matching

We introduce Flux Matching, a new paradigm for generative modeling that generalizes existing score-based models to a broader family of vector fields that need not be conservative. Rather than requiring the model to equal the data score, the Flux Matching objective imposes a weaker condition that admits infinitely many vector fields whose stationary distribution is the data. This flexibility enables a class of generative models that cannot be learned under score matching, in which inductive biases, structural priors, and properties of the dynamics can be directly imposed or optimized. We show that Flux Matching performs strongly on high-dimensional image datasets and, more importantly, that our added freedom unlocks a range of applications including faster sampling, interpretable and mechanistic models, and dynamics that encode directed dependencies between variables. More broadly, Flux Matching opens a new dimension in generative modeling by turning the vector field itself into a design choice rather than a fixed target. Code is available at https://github.com/peterpaohuang/flux_matching.

Peter Pao-Huang, Xiaojie Qiu, Stefano Ermon

Department of Computer Science, Stanford University · Department of Genetics, Stanford University

Generative Modeling Score-Based Generative Modeling

May 7, 2026cs.LG

Kurtosis-Guided Denoising Score Matching for Tabular Anomaly Detection

Denoising score matching (DSM) provides a way to learn data distributions by training a neural network to recover the score function, defined as the gradient of the log density, from noise-corrupted samples. Once trained, the score magnitude at a test point reflects how consistent that point is with the learned distribution, making it a natural anomaly signal. The key practical challenge is selecting the perturbation scale: too little noise yields unstable score estimates in sparse regions, while too much erases local structure and weakens anomaly sensitivity. This is compounded by the difficulty of hyperparameter tuning when anomalies are unknown and no validation set is available. We introduce kurtosis-based noise scaling (K-DSM), a per-feature scheme that sets noise levels from the shape of each marginal distribution, improving coverage of low-density regions and precision in high-density regions without extra model complexity. Contrary to prior claims that multi-scale or noise-conditioned training is necessary, we find that a carefully trained single-scale model is already a strong anomaly detector. On standard tabular anomaly detection benchmarks, K-DSM achieves state-of-the-art performance in the semi-supervised setting. When combined with a lightweight EMA-teacher filtering rule that removes low-density training points before each gradient step, it also achieves strong performance in the fully unsupervised (contaminated) setting, suggesting that simple, data-adaptive noise scaling enables robust anomaly detection while reducing reliance on hyperparameter tuning.

Victor Livernoche, Jie Zan, Reihaneh Rabbany

McGill University, Mila, Montréal, Canada

Tabular Anomaly Detection Unsupervised Anomaly Detection Tabular ML Score-Based Generative Modeling Anomaly Detection

May 7, 2026cs.LG

A Unified Measure-Theoretic View of Diffusion, Score-Based, and Flow Matching Generative Models

We survey continuous-time generative modeling methods based on transporting a simple reference distribution to a data distribution via stochastic or deterministic dynamics. We present a unified framework in which diffusion models, score-based generative models, and flow matching are instances of learning a time-dependent vector field that induces a family of marginals

(ρ_t)_{t \in [0,1]}

governed by continuity and Fokker-Planck equations. Such a unified theory is timely because these methods are converging methodologically, yet fragmented notation and competing derivations continue to obscure their shared structure and the practical tradeoffs governing sampling, stability, and computation. Within this framework, we (i) derive reverse-time sampling for diffusion and score-based models as controlled stochastic dynamics, (ii) show that the probability flow ODE yields identical marginals and connects diffusion to likelihood-based normalizing flows, and (iii) interpret flow matching as direct regression of the velocity field under a chosen interpolation, clarifying when it coincides with or differs from score-based training. We compare objectives, sampling schemes, and discretization errors under unified notation, discuss connections to Schrodinger bridges and entropic optimal transport, and summarize theoretical guarantees and open problems on approximation, stability, and scalability.

Aditya Ranganath, Mukesh Singhal

Center for Applied Scientific Computing Lawrence Livermore National Laboratory Livermore, CA 94551, USA · Department of Electrical Engineering University of California Merced, CA 95343, USA

Flow Matching Generative Modeling Flow-Based Generative Modeling Score-Based Generative Modeling Diffusion Models

May 7, 2026stat.ML

The Interplay of Data Structure and Imbalance in the Learning Dynamics of Diffusion Models

Real-world datasets are inherently heterogeneous, yet how per-class structural differences and sampling imbalance shape the training dynamics of diffusion models-and potentially exacerbate disparities-remains poorly understood. While models typically transition from an initial phase of generalization to memorizing the training set, existing theory assumes homogeneous data, leaving open how class imbalance and heterogeneity reshape these dynamics. In this work, we develop a high-dimensional analytical framework to study class-dependent learning in score-based diffusion models. Analyzing a random-features model trained on Gaussian mixtures, we derive the feature-covariance spectrum to characterize per-class generalization and memorization times. We reveal the explicit hierarchy governing these dynamics: class variance is the primary determinant of learning order-consistently favoring higher-variance classes-while centroid geometry plays a secondary role. Sampling imbalance acts as a modulator that can reverse this ordering and, under strong imbalance, forces minority classes to acquire distinct, delayed speciation times during backward diffusion. Together, these results suggest that diffusion models can memorize some classes while others remain insufficiently learned. We validate our theoretical predictions empirically using U-Net models trained on Fashion MNIST.

Flavio Nicoletti, Chenxiao Ma, Enrico Ventura +2

Data Science and AI, Computer Science and Engineering Department, Chalmers University of Technology and University of Gothenburg · International School of Advanced Studies (SISSA), Trieste, Italy · Department of Computing Sciences, Bocconi University, Milano, Italy +2

Class-Imbalanced Learning Score-Based Generative Modeling Diffusion Models

May 7, 2026eess.AS

Predictive-Generative Drift Decomposition for Speech Enhancement and Separation

We propose a plug-and-play framework for speech enhancement and separation that augments predictive methods with a generative speech prior. Our approach, termed Stochastic Interpolant Prior for Speech (SIPS), builds on stochastic interpolants and leverages their flexibility to bridge predictive and generative modeling. Specifically, we decompose the interpolation dynamics into a task-specific drift and a stochastic denoising component, allowing a predictive estimate to be integrated directly into the generative sampling process. This results in a mathematically grounded framework for combining strong pretrained predictors with the expressive power of generative models. To this end, we train a score model using only clean speech, yielding a degradation-agnostic prior that can be reused across tasks. During inference, the predictor provides a deterministic drift that steers the sampling process toward a task-consistent estimate, while the score model preserves perceptual naturalness. Unlike prior hybrid approaches, which typically rely on architecture-specific conditioning and are tied to particular predictors or degradation settings, SIPS provides a unified framework that generalizes across predictors and additive degradation tasks. We demonstrate its effectiveness for both speech enhancement and speech separation using recent predictors such as SEMamba and FlexIO. The proposed method consistently improves perceptual quality, achieving gains up +1.0 NISQA for speech separation.

Julius Richter, Yoshiki Masuyama, Christoph Boeddeker +3

MERL Cambridge, MA, USA

Stochastic Interpolants Speech Processing Speech Enhancement Score-Based Generative Modeling Speech Separation

May 5, 2026stat.ML

Stochastic Schrödinger Diffusion Models for Pure-State Ensemble Generation

Quantum machine learning increasingly relies on pure-state representations, motivating generative models that sample directly in quantum representation space rather than perturbing classical inputs and re-encoding. We introduce Stochastic Schrödinger Diffusion Models (SSDMs), a score-based generative framework that defines diffusion, scores, and reverse-time sampling intrinsically on the complex projective manifold

\mathbb{CP}^{d-1}

under the Fubini--Study metric. SSDMs combine a Riemannian Ornstein--Uhlenbeck forward diffusion with a stochastic Schrödinger realization, and learn reverse-time dynamics driven by the Riemannian score. Our central technical contribution is a local-time learning objective that exploits the local Euclidean OU limit of intrinsic manifold diffusions in Fubini-Study normal coordinates to obtain an analytic teacher score, bypassing the intractable transition densities that limit existing Riemannian score-based models. Across synthetic, physics-inspired (TFIM, XXZ), and quantum feature-state benchmarks up to

14

qubits, SSDMs match target pure-state ensembles by orders of magnitude on MMD and observable statistics over both ambient Euclidean and matched Riemannian score-based baselines, and improve representation-level diagnostics for downstream quantum kernel methods.

Jian Xu, Wei Chen, Shigui Li +5

1RIKEN iTHEMS · 2RIKEN AIP · 3South China University of Technology +2

Quantum Generative Modeling Quantum State Preparation Score-Based Generative Modeling Diffusion Models

Apr 30, 2026stat.ML

A unified perspective on fine-tuning and sampling with diffusion and flow models

We study the problem of training diffusion and flow generative models to sample from target distributions defined by an exponential tilting of a base density; a formulation that subsumes both sampling from unnormalized densities and reward fine-tuning of pre-trained models. This problem can be approached from a stochastic optimal control (SOC) perspective, using adjoint-based or score matching methods, or from a non-equilibrium thermodynamics perspective. We provide a unified framework encompassing these approaches and make three main contributions: (i) bias-variance decompositions revealing that Adjoint Matching/Sampling and Novel Score Matching have finite gradient variance, while Target and Conditional Score Matching do not; (ii) norm bounds on the lean adjoint ODE that theoretically support the effectiveness of adjoint-based methods; and (iii) adaptations of the CMCD and NETS loss functions, along with novel Crooks and Jarzynski identities, to the exponential tilting setting. We validate our analysis with reward fine-tuning experiments on Stable Diffusion 1.5 and 3.

Carles Domingo-Enrich, Yuanqi Du, Michael S. Albergo

Microsoft Research New England · Harvard University, Kempner Institute

Stochastic Optimal Control Diffusion Sampling Diffusion Model Fine-Tuning Flow-Based Generative Modeling Score-Based Generative Modeling

Apr 28, 2026quant-ph

Quantum Dynamics via Score Matching on Bohmian Trajectories

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.

Lei Wang

Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China

Score Matching PDE Solving Score-Based Generative Modeling

Apr 26, 2026cs.LG

Symmetric Equilibrium Propagation for Thermodynamic Diffusion Training

The reverse process in score-based diffusion models is formally equivalent to overdamped Langevin dynamics in a time-dependent energy landscape. In our prior work we showed that a bilinearly-coupled analog substrate can physically realize this dynamics at a projected three-to-four orders of magnitude energy advantage over digital inference by replacing dense skip connections with low-rank inter-module couplings. Whether the \emph{training} loop can be closed on the same substrate -- without routing gradients through an external digital accelerator -- has remained open. We resolve this affirmatively: Equilibrium Propagation applied directly to the bilinear energy yields an unbiased estimator of the denoising score-matching gradient in the zero-nudge limit. For finite nudging we derive a sharp bias bound controlled solely by substrate stiffness, local curvature, and the norm of the loss-gradient signal, with a bilinear-specific corollary showing that one dominant bias term vanishes identically for coupling-parameter updates. Symmetric nudging further upgrades the leading bias from

\mathcal{O}(β)

to

\mathcal{O}(β^2)

at negligible extra cost. Under realistic finite-relaxation budgets this upgrade is essential, as one-sided EqProp produces anti-correlated gradients while symmetric EqProp yields well-aligned updates. Bias-variance analysis determines the optimal operating point, and end-to-end physical-unit accounting projects a

10^3

-

10^4\times

energy advantage per training step over a matched GPU baseline. Symmetric bilinear EqProp is the first local, readout-only training rule that preserves the low-rank coupling enabling scalable thermodynamic diffusion models.

Aditi De

Equilibrium Propagation Thermodynamic Computing Score-Based Generative Modeling Diffusion Models

Apr 20, 2026cs.CV

Score-Based Matching with Target Guidance for Cryo-EM Denoising

Cryo-electron microscopy (cryo-EM) enables single-particle analysis of biological macromolecules under strict low-dose imaging conditions, but the resulting micrographs often exhibit extremely low signal-to-noise ratios and weak particle visibility. Image denoising is therefore an important preprocessing step for downstream cryo-EM analysis, including particle picking, 2D classification, and 3D reconstruction. Existing cryo-EM denoising methods are commonly trained with pixel-wise or Noise2Noise-style objectives, which can improve visual quality but do not explicitly account for structural consistency required by downstream analysis. In this work, we propose a score-based denoising framework for cryo-EM that learns the clean-data score to recover particle signals while better preserving structural information. Building on this formulation, we further introduce a target-guided variant that incorporates reference-density guidance to stabilize score learning under weak and ambiguous signal conditions. Rather than simply amplifying particle-like responses, our framework better suppresses structured low-frequency background, which improves particle--background separability for downstream analysis. Experiments on multiple cryo-EM datasets show that our score-based methods consistently improve downstream particle picking and produce more structure-consistent 3D reconstructions. Experiments on multiple cryo-EM datasets show that our methods improve downstream particle picking and produce more structure-consistent reconstructions.

Xiaoqi Wu, Xueying Zhan, Wen Li +3

Carnegie Mellon University · Towson University

Cryo-Electron Microscopy Score-Based Generative Modeling Image Denoising

Apr 19, 2026cs.LG

Reward Score Matching: Unifying Reward-based Fine-tuning for Flow and Diffusion Models

Reward-based fine-tuning steers a pretrained diffusion or flow-based generative model toward higher-reward samples while remaining close to the pretrained model. Although existing methods are derived from different perspectives, we show that many can be written under a common framework, which we call reward score matching (RSM). Under this view, alignment becomes score matching against a value-guided target, and the main differences across methods reduce to the construction of the value-guidance estimator and the effective optimization strength across timesteps. This unification clarifies the bias-variance-compute tradeoffs of existing designs, and distinguishes core optimization components from auxiliary mechanisms that add complexity without clear benefit. Guided by this perspective, we develop simpler, more efficient redesigns across representative differentiable and black-box reward alignment tasks. Overall, RSM turns a seemingly fragmented collection of reward-based fine-tuning methods into a smaller, more interpretable, and more actionable design space. Code is available at https://github.com/jaylee2000/rsm

Jeongjae Lee, Jinho Chang, Jeongsol Kim +1

Graduate School of AI KAIST, Korea

Diffusion Model Alignment Diffusion Model Fine-Tuning Score-Based Generative Modeling

Apr 16, 2026cs.CV

An Analysis of Regularization and Fokker-Planck Residuals in Diffusion Models for Image Generation

Recent work has shown that diffusion models trained with the denoising score matching (DSM) objective often violate the Fokker--Planck (FP) equation that governs the evolution of the true data density. Directly penalizing these deviations in the objective function reduces their magnitude but introduces a significant computational overhead. It is also observed that enforcing strict adherence to the FP equation does not necessarily lead to improvements in the quality of the generated samples, as often the best results are obtained with weaker FP regularization. In this paper, we investigate whether simpler penalty terms can provide similar benefits. We empirically analyze several lightweight regularizers, study their effect on FP residuals and generation quality, and show that the benefits of FP regularization are available at substantially lower computational cost. Our code is available at https://github.com/OnnoNiemann/fp_diffusion_analysis.

Onno Niemann, Gonzalo Martínez Muñoz, Alberto Suárez Gonzalez

Universidad Autónoma de Madrid

Diffusion-Based Image Generation Score-Based Generative Modeling Diffusion Models

Mar 24, 2026cs.LG

Asymptotic Learning Curves for Diffusion Models with Random Features Score and Manifold Data

We study the theoretical behavior of denoising score matching--the learning task associated to diffusion models--when the data distribution is supported on a low-dimensional manifold and the score is parameterized using a random feature neural network. We derive asymptotically exact expressions for the test, train, and score errors in the high-dimensional limit. Our analysis reveals that, for linear manifolds the sample complexity required to learn the score function scales linearly with the intrinsic dimension of the manifold, rather than with the ambient dimension. Perhaps surprisingly, the benefits of low-dimensional structure starts to diminish once we have a non-linear manifold. These results indicate that diffusion models can benefit from structured data; however, the dependence on the specific type of structure is subtle and intricate.

Anand Jerry George, Nicolas Macris

École Polytechnique Fédérale de Lausanne (EPFL), Lab for Statistical Mechanics of Inference in Large Systems (SMILS), CH-1015 Lausanne, Switzerland

Intrinsic Dimensionality Score-Based Generative Modeling Statistical Learning Theory Diffusion Models