arXiv:2512.01820v1 Announce Type: new 
Abstract: Diffusion generative models have emerged as powerful tools for producing synthetic data from an empirically observed distribution. A common approach involves simulating the time-reversal of an Ornstein-Uhlenbeck (OU) process initialized at the true data distribution. Since the score function associated with the OU process is typically unknown, it is approximated using a trained neural network. This approximation, along with finite time simulation, time discretization and statistical approximation, introduce several sources of error whose impact on the generated samples must be carefully understood. Previous analyses have quantified the error between the generated and the true data distributions in terms of Wasserstein distance or Kullback-Leibler (KL) divergence. However, both metrics present limitations: KL divergence requires absolute continuity between distributions, while Wasserstein distance, though more general, leads to error bounds that scale poorly with dimension, rendering them impractical in high-dimensional settings. In this work, we derive an explicit, dimension-free bound on the discrepancy between the generated and the true data distributions. The bound is expressed in terms of a smooth test functional with bounded first and second derivatives. The key novelty lies in the use of this weaker, functional metric to obtain dimension-independent guarantees, at the cost of higher regularity on the test functions. As an application, we formulate and solve a variational problem to minimize the time-discretization error, leading to the derivation of an optimal time-scheduling strategy for the reverse-time diffusion. Interestingly, this scheduler has appeared previously in the literature in a different context; our analysis provides a new justification for its optimality, now grounded in minimizing the discretization bias in generative sampling.

تقدم دراسة جديدة تقديرًا للخطأ بدون أبعاد لنماذج التوليد الانتشاري، التي تُستخدم بشكل متزايد لإنتاج بيانات اصطناعية من توزيعات العالم الحقيقي. تسلط الأبحاث الضوء على التحديات التي تطرحها تقريب دالة الدرجات لعملية أورنشتاين-أوهلنبيك، مع التأكيد على الأخطاء التي يتم إدخالها من خلال التمييز الزمني والتقريبات الإحصائية.

Un nuevo estudio presenta una estimación de error sin dimensiones para los modelos generativos de difusión, que se utilizan cada vez más para generar datos sintéticos a partir de distribuciones del mundo real. La investigación destaca los desafíos que plantea la aproximación de la función de puntuación del proceso de Ornstein-Uhlenbeck, enfatizando los errores introducidos a través de la discretización temporal y las aproximaciones estadísticas.

Une nouvelle étude présente une estimation d'erreur sans dimension pour les modèles génératifs de diffusion, de plus en plus utilisés pour générer des données synthétiques à partir de distributions réelles. La recherche met en lumière les défis posés par l'approximation de la fonction de score du processus d'Ornstein-Uhlenbeck, en soulignant les erreurs introduites par la discrétisation temporelle et les approximations statistiques.

A new study presents a dimension-free error estimate for diffusion generative models, which are increasingly utilized for generating synthetic data from real-world distributions. The research highlights the challenges posed by approximating the score function of the Ornstein-Uhlenbeck process, emphasizing the errors introduced through time discretization and statistical approximations.

Dimension-free error estimate for diffusion model and optimal scheduling

arXiv:2505.18276v2 Announce Type: replace 
Abstract: Designing algorithms for solving high-dimensional Bayesian inverse problems directly in infinite-dimensional function spaces - where such problems are naturally formulated - is crucial to ensure stability and convergence as the discretization of the underlying problem is refined. In this paper, we contribute to this line of work by analyzing a widely used sampler for linear inverse problems: Langevin dynamics driven by score-based generative models (SGMs) acting as priors, formulated directly in function space. Building on the theoretical framework for SGMs in Hilbert spaces, we give a rigorous definition of this sampler in the infinite-dimensional setting and derive, for the first time, error estimates that explicitly depend on the approximation error of the score. As a consequence, we obtain sufficient conditions for global convergence in Kullback-Leibler divergence on the underlying function space. Preventing numerical instabilities requires preconditioning of the Langevin algorithm and we prove the existence and the form of an optimal preconditioner. The preconditioner depends on both the score error and the forward operator and guarantees a uniform convergence rate across all posterior modes. Our analysis applies to both Gaussian and a general class of non-Gaussian priors. Finally, we present examples that illustrate and validate our theoretical findings.

قدمت دراسة جديدة إطارًا صارمًا لتطبيق ديناميات لانجفين المدفوعة بنماذج توليدية قائمة على الدرجات (SGMs) على مشاكل عكسية بايزية خطية في أبعاد لانهائية، موفرةً تقديرات للخطأ تعتمد على أخطاء تقريب الدرجات. تهدف هذه التطورات إلى تعزيز الاستقرار والتقارب في المشاكل البايزية عالية الأبعاد من خلال تعريف العينات مباشرة في الفضاء الوظيفي.

Un nuevo estudio ha introducido un marco riguroso para aplicar la dinámica de Langevin impulsada por modelos generativos basados en puntajes (SGMs) a problemas inversos bayesianos lineales en dimensiones infinitas, proporcionando estimaciones de error que dependen de los errores de aproximación de puntajes. Este avance busca mejorar la estabilidad y la convergencia en problemas bayesianos de alta dimensión al definir muestreadores directamente en el espacio funcional.

Une nouvelle étude a introduit un cadre rigoureux pour appliquer la dynamique de Langevin, guidée par des modèles génératifs basés sur les scores (SGMs), à des problèmes inverses bayésiens linéaires en dimensions infinies, fournissant des estimations d'erreur dépendant des erreurs d'approximation des scores. Cette avancée vise à améliorer la stabilité et la convergence dans des problèmes bayésiens de haute dimension en définissant des échantillonneurs directement dans l'espace fonctionnel.

A new study has introduced a rigorous framework for applying Langevin dynamics driven by score-based generative models (SGMs) to infinite-dimensional linear Bayesian inverse problems, providing error estimates that depend on score approximation errors. This advancement aims to enhance stability and convergence in high-dimensional Bayesian problems by defining samplers directly in function space.

Dimension-free error estimate for diffusion model and optimal scheduling

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