arXiv:2511.12881v1 Announce Type: cross 
Abstract: Leveraging the Wasserstein distance -- a summation of sample-wise transport distances in data space -- is advantageous in many applications for measuring support differences between two underlying density functions. However, when supports significantly overlap while densities exhibit substantial pointwise differences, it remains unclear whether and how this transport information can accurately identify these differences, particularly their analytic characterization in finite-sample settings. We address this issue by conducting an analysis of the information processing capabilities of the one-dimensional Wasserstein distance with finite samples. By utilizing the Poisson process and isolating the rate factor, we demonstrate the capability of capturing the pointwise density difference with Wasserstein distances and how this information harmonizes with support differences. The analyzed properties are confirmed using neural spike train decoding and amino acid contact frequency data. The results reveal that the one-dimensional Wasserstein distance highlights meaningful density differences related to both rate and support.

يتناول المقال مزايا استخدام مسافة فاسرشتاين لقياس الفروق بين دالتين كثافيتين، خاصة في الحالات التي تتداخل فيها دعائمهما بشكل كبير. يبرز التحديات في تحديد الفروق الدقيقة بدقة في إعدادات العينات المحدودة. يقوم المؤلفون بتحليل قدرات معالجة المعلومات لمسافات فاسرشتاين أحادية البعد باستخدام عملية بواسون وعزل عامل المعدل. تظهر نتائجهم قدرة مسافات فاسرشتاين على التقاط الفروق الدقيقة في الكثافة وكيف يرتبط ذلك بفروق الدعم، مما تم تأكيده من خلال فك تشفير سلسلة النبضات العصبية.

El artículo discute las ventajas de utilizar la distancia de Wasserstein para medir las diferencias entre dos funciones de densidad, especialmente en casos donde sus soportes se superponen significativamente. Destaca los desafíos para identificar con precisión las diferencias de densidad puntual en configuraciones de muestras finitas. Los autores analizan las capacidades de procesamiento de información de las distancias de Wasserstein unidimensionales utilizando el proceso de Poisson y aislando el factor de tasa. Sus hallazgos demuestran la capacidad de las distancias de Wasserstein para captu…

L'article traite des avantages de l'utilisation de la distance de Wasserstein pour mesurer les différences entre deux fonctions de densité, en particulier dans les cas où leurs supports se chevauchent considérablement. Il met en évidence les défis liés à l'identification précise des différences de densité ponctuelles dans des contextes d'échantillons finis. Les auteurs analysent les capacités de traitement de l'information des distances de Wasserstein unidimensionnelles en utilisant le processus de Poisson et en isolant le facteur de taux. Leurs résultats démontrent la capacité des distances d…

The article discusses the advantages of using the Wasserstein distance to measure differences between two density functions, particularly in cases where their supports overlap significantly. It highlights the challenges in accurately identifying pointwise density differences in finite sample settings. The authors analyze the information processing capabilities of one-dimensional Wasserstein distances by employing the Poisson process and isolating the rate factor. Their findings demonstrate the ability of Wasserstein distances to capture pointwise density differences and how this relates to sup…

On the Information Processing of One-Dimensional Wasserstein Distances with Finite Samples

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.

On the Information Processing of One-Dimensional Wasserstein Distances with Finite Samples

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