Score Matching

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Score matching fits a model by matching its score—the gradient of log density—to data-derived or denoising targets, avoiding the need to evaluate a density’s normalizing constant. Its main families include classical and generalized score matching for exponential families and constrained domains, denoising score matching across noise scales in diffusion models, and methods adapted to manifolds or discrete data. These tools support generative sampling and also appear in causal discovery, inverse problems, scientific simulation, and statistical estimation. A central challenge is translating score accuracy into reliable outcomes: finite-sample error, optimization, boundary behavior, and reverse-diffusion stability can affect generated samples, while standard score-loss measures may not capture those effects.

Latest papers

36 papers9 papers in the last four weeks, up 200% on the four weeks before. 0.1% of all new papers.

Approximation Families and the Statistical Efficiency of Denoising Score Matching
Oct 26, 2026Brian MiyatakeScore Matching Score-Based Generative Modeling
Department of Statistics and Data Science, Yale University
Learning the Autoregressive: Direct Score-Difference Estimation with Amortized Future Credit for Video Distillation
Oct 25, 2026Zixuan Duan, Yabo Chen, Xunzhi Xiang +5Autoregressive Video Generation Diffusion Model Distillation
Nanjing University · Institute of Artificial Intelligence, China Telecom (TeleAI) · Fudan University
A unified approach for estimation of unnormalized models via Stein operators
Oct 21, 2026Mitsuki Nagai, Keisuke YanoParameter Estimation Proper Scoring Rules
Graduate Institute for Advanced Studies, SOKENDAI · The Institute of Statistical Mathematics
StochLOB: Heavy-Tailed Score Matching for Robust Limit Order Book Trend Prediction
Oct 15, 2026Arshia Abolghasemi, Seyed Reza Tavakoli, Parsa Naderi +7Limit Order Book Forecasting Score Matching
Dataism Laboratory for Quantitative Finance · Sharif University of Technology · Amirkabir University of Technology +3
Score Mismatch and Set-Constrained Sampling in Zero-Shot Conditional Discrete Diffusion
Oct 15, 2026Renxiang Huang, Yuchen Liang, Yingbin Liang +2Diffusion Models Diffusion Sampling
Department of Electrical and Computer Engineering, University of California, Davis, Davis, CA, 95616, USA · Luddy School of Informatics, Computing, and Engineering, Indiana University Indianapolis, Indianapolis, IN, 46202, USA · Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH, 43210, USA
Data Free Training of Diffusion Models for Lattice Field Theories
Oct 13, 2026Dennis Grinwald, Ankur Singha, Khaled Kahouli +2Lattice Field Theory Diffusion Model Sampling
Berlin Institute for the Foundations of Learning and Data (BIFOLD), Germany · Technische Universität Berlin, Germany · Deutsches Elektronen-Synchrotron (DESY), Platanenallee 6, 15738 Zeuthen, Germany +2
Causal Discovery Under Hard Selection Bias: A New Robust Score-Matching Approach
Oct 13, 2026Yiwen Qiu, Francesco Montagna, Shimeng Huang +1Causal Discovery Selection Bias
IST Austria, Am Campus 1, Klosterneuburg, 3400
A Stochastic Analysis Perspective on Diffusion Models
Oct 9, 2026Stanislas StrasmanScore-Based Generative Modeling Diffusion Sampling
SU, LPSM · Laboratoire de Probabilités, Statistique et Modélisation, Sorbonne Université
Latent Score-Based Bayesian Cramér-Rao Bound Estimation for High-Dimensional Imaging Systems
Oct 2, 2026Evan Scope Crafts, Thomas Wynn, Seonyeong Park +2Photoacoustic Tomography Bayesian Inverse Problems
Oden Institute for Computational Engineering and Sciences, the University of Texas at Austin, Austin, TX, 78712 · Graduate School of Biomedical Sciences at UT Southwestern Medical Center · Mallinckrodt Institute of Radiology at Washington University in St. Louis +1
Discrete Score Matching Enables Causal Discovery from Count Data
Sep 30, 2026Euijong Song, Hyewon Park, Gunwoong ParkScore Matching Causal Discovery
Department of Statistics Seoul National University, Korea · Interdisciplinary Program in Artificial Intelligence; Institute for Data Innovation in Science, Seoul National University, Korea
Safe Score Matching: Diffusion Policies with Hamilton-Jacobi Reachability for Online Safe Reinforcement Learning
Sep 27, 2026Boyang Li, Matthew Kim, Sylvia Lee HerbertScore Matching Constrained RL
University of California San Diego
Generalised Score Matching on Convex Domains
Sep 10, 2026Nishanth Shetty, Saisuchith Mahajan, Chandra Sekhar SeelamantulaScore Matching Parameter Estimation
Department of Electrical Engineering Indian Institute of Science, Bengaluru 560012
Connecting Score Matching, Maximum Likelihood, and Expectation-Maximization in Mixed Linear Regression
Sep 4, 2026Zhankun Luo, Abolfazl HashemiScore Matching Mixed-Effects Models
School of Electrical and Computer Engineering Purdue University, West Lafayette, IN, USA
Sobolev Regularized Score Difference Estimation in Diffusion Models
Aug 18, 2026Chenghan Xie, Jose Blanchet, Renyuan XuDiffusion Models Score Matching
Department of Management Science and Engineering, Stanford University.
A Dual Evaluation for Music Transcription
Aug 5, 2026Ping Wang, Guang Yang, Nazif Can Tamer +2Score Matching Automatic Music Transcription
Should the Boundary Term Be Learned in Reflected Diffusion? Conormal Trace and Reflection Masking
Aug 4, 2026Ziyue Wang, Takafumi KanamoriScore Matching Diffusion Models
Institute of Science Tokyo
A Hyperfinite Framework for Score-Based Generative Modeling
Aug 3, 2026Sunder Ram KrishnanScore Matching Stochastic Differential Equations
Kerala School of Mathematics, Kozhikode 673571, Kerala, India.
Score Accuracy Along the Forward Diffusion Does Not Certify Numerical Stability in Diffusion Sampling
Jul 9, 2026Yiwei ZhouScore Matching Diffusion Sampling
Tightening the Score Matching Gap for Diffusion Models
Jul 5, 2026Benjamin Dupuis, Tyler Farghly, Maxime Haddouche +2Score Matching Score-Based Generative Modeling
INRIA - CNRS - D´epartement d’Informatique de l’Ecole Normale Sup´erieure - PSL Research University, France · Ecole Polytechnique - CMAP, IP Paris, Palaiseau, France
Global Convergence of Gradient Descent for Score Matching in Gaussian Mixtures via Reverse Fisher Divergence
Jun 18, 2026Alexander TyurinScore Matching Optimization Convergence Analysis
AXXX, Moscow, Russia · Applied AI Institute, Moscow, Russia
Diffusion Models Observe Only Gradients: A Geometric Perspective on Score Matching Errors
Jun 4, 2026Naïl B. Khelifa, Richard E. Turner, Ramji VenkataramananScore Matching Hodge Decomposition
University of Cambridge
Torus Graphs for Large Scale Neural Phase Analysis
May 30, 2026Jack Goffinet, Casey Hanks, David E. CarlsonScore Matching Sequence Modeling
Department of Computer Science, Duke University, Durham NC, USA · Departments of Civil and Environmental Engineering; Biostatistics and Bioinformatics; and Electrical and Computer Engineering, Duke University, Durham NC, USA
Score-Control for Hallucination Reduction in Diffusion Models
May 29, 2026Mahesh Bhosale, Naresh Kumar Devulapally, Abdul Wasi +3Score Matching Diffusion-Based Image Generation
University at Buffalo Buffalo, NY, USA
Rao-Blackwellized Score Matching on Manifolds
May 25, 2026Divit RawalScore Matching Score-Based Generative Modeling
Department of Statistics, University of California, Berkeley, Berkeley CA, U.S.A..
Diffusion-based Denoising Beats Vanilla Score Matching in Parameter Estimation: A Theoretical Explanation
May 21, 2026Benedikt Lütke Schwienhorst, Nadja Klein, Johannes LedererScore Matching Parameter Estimation
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
Finite Sample Bounds for Learning with Score Matching
May 13, 2026Devin Smedira, Abhijith Jayakumar, Sidhant Misra +2Score Matching Sample Complexity
Operations Research Center, Massachusetts Institute of Technology · Theoretical division, Los Alamos National Lab
Proximal-Based Generative Modeling for Bayesian Inverse Problems
May 13, 2026Boyang Zhang, Zhiguo Wang, Ya-Feng LiuScore Matching Bayesian Inverse Problems
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
Diffusion model for SU(N) gauge theories
May 7, 2026Javad Komijani, Marina K. Marinkovic, Lara TurgutScore Matching Diffusion Sampling
Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland
Quantum Dynamics via Score Matching on Bohmian Trajectories
Apr 28, 2026Lei WangScore Matching PDE Solving
Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
Transformers Learn the Optimal DDPM Denoiser for Multi-Token GMMs
Apr 11, 2026Hongkang Li, Hancheng Min, Rene VidalDiffusion Models Diffusion-Based Denoising
Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, USA · Institute of Natural Sciences & School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China

Oct 26, 2026J/K move · Enter open · S save

Approximation Families and the Statistical Efficiency of Denoising Score Matching

Brian Miyatake

Department of Statistics and Data Science, Yale University

Score Matching Score-Based Generative Modeling Density Estimation Minimax Estimation

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We study score matching over finite-dimensional approximation spaces generated by gradients of eigenfunctions of the infinitesimal generator of a forward diffusion. We compare polynomial approximation under one-dimensional Ornstein-Uhlenbeck flow with trigonometric-polynomial approximation under heat flow on the torus. In the Ornstein-Uhlenbeck setting, we show that for every non-Gaussian initial distribution uniformly comparable to the standard Gaussian, the asymptotic score-estimation variance grows exponentially along a subsequence of polynomial degrees. Averaging the score-matching loss under the diffusion semigroup, which we call operator-informed score matching (OISM), reduces the variance at each fixed degree but does not eliminate this exponential instability at sufficiently small times. A local minimax lower bound shows that the same high-degree dependence arises when estimating the polynomial projection of the score, rather than being an artifact of OISM. In contrast, under heat flow on the torus, OISM over trigonometric polynomials admits finite-sample control of the score error along the forward diffusion and yields the classical minimax rate for estimating sufficiently smooth densities under expected Kullback-Leibler loss.