arXiv:2605.28625v1 Announce Type: new 
Abstract: Generative modeling provides a powerful framework for learning data distributions. These models initially relied on probabilistic methods such as Gaussian Processes (GP) for uncertainty-aware predictions and shifted towards larger trainable models to learn more complex distributions. In this work, we introduce Random Process (RP) Flow, a Flow Matching-based framework that represents the vector field as a neural implicit function. Unlike modern generative methods, our setting involves a single observed field, from which only sparse measurements are available. RP Flow uses Random Fourier Features to learn an implicit signal representation that can be queried at any arbitrary location from a limited set of observations, while encoding uncertainty through ensemble sampling. We propose constructing a Bayesian posterior by GP regression in the source space to generate high-quality samples. Our empirical results demonstrate that this framework generates realistic samples along with calibrated uncertainty estimates, even under challenging conditions such as high frequency, high sparsity, or high dimensionality. These findings position RP Flow as a milestone towards generative models for reconstruction tasks where data is scarce and uncertainty must remain traceable.

تم تقديم إطار عمل جديد يسمى Random Process Flow، الذي يستخدم Flow Matching لتمثيل الحقول المتجهة كوظائف ضمنية عصبية. تتيح هذه الطريقة تعلم توزيعات بيانات معقدة من ملاحظات نادرة، مستفيدة من ميزات فورييه العشوائية لإنشاء تمثيل إشارة ضمني بينما يتم ترميز عدم اليقين من خلال أخذ عينات جماعية.

Se ha introducido un nuevo marco llamado Random Process Flow, que utiliza el Flow Matching para representar campos vectoriales como funciones implícitas neuronales. Este enfoque permite aprender distribuciones de datos complejas a partir de observaciones escasas, aprovechando las características de Fourier aleatorias para crear una representación de señal implícita mientras se codifica la incertidumbre a través del muestreo en conjunto.

Un nouveau cadre appelé Random Process Flow a été introduit, utilisant le Flow Matching pour représenter les champs vectoriels comme des fonctions implicites neuronales. Cette approche permet d'apprendre des distributions de données complexes à partir d'observations rares, en s'appuyant sur des caractéristiques de Fourier aléatoires pour créer une représentation de signal implicite tout en encodant l'incertitude par le biais d'un échantillonnage en ensemble.

A new framework called Random Process Flow has been introduced, which utilizes Flow Matching to represent vector fields as neural implicit functions. This approach allows for the learning of complex data distributions from sparse observations, leveraging Random Fourier Features to create an implicit signal representation while encoding uncertainty through ensemble sampling.

Random Process Flow Matching: Generative Implicit Representations of Multivariate Random Fields

arXiv:2505.24066v2 Announce Type: replace-cross 
Abstract: Finite-rank approximations are widely used to scale Gaussian process (GP) regression, but their posterior behavior can differ from that of the corresponding parent GP prior. We study a class of finite-rank GP priors built from locally supported basis expansions with dependent Gaussian coefficients. Our framework covers finite-element approximations based on the stochastic partial differential equation (SPDE) representation of Mat\'ern GPs and regular-grid GP interpolation schemes. We show that, with a suitable prior on the resolution parameter $N$, these finite-rank expansions inherit the same posterior contraction rate as the corresponding parent GP prior under the same bandwidth specification used for that parent prior. Consequently, the interpolation construction under a squared-exponential parent GP attains the minimax-optimal rate up to logarithmic factors under a hierarchical prior on the bandwidth parameter and on $N$, while the SPDE construction attains the same rate under a bandwidth scaling depending on the sample size and the smoothness of the true function, together with a prior on $N$. We also develop a posterior sampler for the hierarchical interpolation model that jointly updates the resolution and bandwidth parameters, and we provide numerical studies that support the theory.

قدمت دراسة حديثة إطارًا لعمليات غاوسي (GP) ذات الرتبة المحدودة تستخدم توسعات أساسية مدعومة محليًا مع معاملات غاوسية معتمدة. تُظهر هذه الطريقة أن التوسعات ذات الرتبة المحدودة يمكن أن تحقق نفس معدل الانكماش البعدي مثل أولوياتها GP الأصلية، خاصةً تحت مواصفات عرض النطاق الترددي المحددة.

Un estudio reciente ha presentado un marco para procesos gaussianos (GP) de rango finito que utiliza expansiones de base localmente soportadas con coeficientes gaussianos dependientes. Este enfoque demuestra que las expansiones de rango finito pueden alcanzar la misma tasa de contracción posterior que sus priors GP correspondientes, especialmente bajo especificaciones de ancho de banda específicas.

Une étude récente a introduit un cadre pour les processus gaussiens (GP) de rang fini qui utilise des expansions de base localement supportées avec des coefficients gaussiens dépendants. Cette approche démontre que les expansions de rang fini peuvent atteindre le même taux de contraction postérieur que leurs priors GP parent, en particulier sous des spécifications de bande passante spécifiques.

A recent study has introduced a framework for finite-rank Gaussian processes (GPs) that utilizes locally supported basis expansions with dependent Gaussian coefficients. This approach demonstrates that finite-rank expansions can achieve the same posterior contraction rate as their parent GP priors, particularly under specific bandwidth specifications.

Random Process Flow Matching: Generative Implicit Representations of Multivariate Random Fields

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