arXiv:2501.02298v5 Announce Type: replace 
Abstract: Score-based Generative Models (SGMs) aim to sample from a target distribution by learning score functions using samples perturbed by Gaussian noise. Existing convergence bounds for SGMs in the W2-distance rely on stringent assumptions about the data distribution. In this work, we present a novel framework for analyzing W2-convergence in SGMs, significantly relaxing traditional assumptions such as log-concavity and score regularity. Leveraging the regularization properties of the Ornstein--Uhlenbeck (OU) process, we show that weak log-concavity of the data distribution evolves into log-concavity over time. This transition is rigorously quantified through a PDE-based analysis of the Hamilton--Jacobi--Bellman equation governing the log-density of the forward process. Moreover, we establish that the drift of the time-reversed OU process alternates between contractive and non-contractive regimes, reflecting the dynamics of concavity. Our approach circumvents the need for stringent regularity conditions on the score function and its estimators, relying instead on milder, more practical assumptions. We demonstrate the wide applicability of this framework through explicit computations on Gaussian mixture models, illustrating its versatility and potential for broader classes of data distributions.

تم تقديم إطار جديد لتحليل التقارب W2 في النماذج التوليدية المعتمدة على الدرجات (SGMs)، والتي كانت تعتمد تقليديًا على افتراضات صارمة مثل اللوغاريتمية والتناسق في الدرجات. يستخدم هذا النهج الجديد عملية أورنشتاين-أوهلنبيك لإظهار أن اللوغاريتمية الضعيفة يمكن أن تتطور إلى لوغاريتمية مع مرور الوقت، مما يوفر فهمًا أكثر مرونة لديناميات التقارب.

Se ha introducido un nuevo marco para analizar la convergencia W2 en los Modelos Generativos Basados en Puntajes (SGMs), que tradicionalmente dependían de suposiciones estrictas como la log-concavidad y la regularidad del puntaje. Este nuevo enfoque utiliza el proceso de Ornstein-Uhlenbeck para demostrar que la débil log-concavidad puede evolucionar hacia la log-concavidad con el tiempo, proporcionando una comprensión más flexible de las dinámicas de convergencia.

Un nouveau cadre a été introduit pour analyser la convergence W2 dans les modèles génératifs basés sur les scores (SGMs), qui reposaient traditionnellement sur des hypothèses strictes telles que la log-concavité et la régularité des scores. Cette nouvelle approche utilise le processus d'Ornstein-Uhlenbeck pour démontrer que la faible log-concavité peut évoluer en log-concavité au fil du temps, offrant une compréhension plus flexible des dynamiques de convergence.

A novel framework has been introduced to analyze W2-convergence in Score-based Generative Models (SGMs), which traditionally relied on strict assumptions like log-concavity and score regularity. This new approach utilizes the Ornstein-Uhlenbeck process to demonstrate that weak log-concavity can evolve into log-concavity over time, providing a more flexible understanding of convergence dynamics.

Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance

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