arXiv:2603.13311v1 Announce Type: new 
Abstract: Multivariate function approximation is a fundamental problem in machine learning. Classic multivariate function approximations rely on hand-crafted basis functions (e.g., polynomial basis and Fourier basis), which limits their approximation ability and data adaptation ability, resulting in unsatisfactory performance. To address these challenges, we introduce the neural basis function by leveraging an untrained neural network as the basis function. Equipped with the proposed neural basis function, we suggest the neural approximation (NeuApprox) paradigm for multivariate function approximation. Specifically, the underlying multivariate function behind the multi-dimensional data is decomposed into a sum of block terms. The clear physically-interpreted block term is the product of expressive neural basis functions and their corresponding learnable coefficients, which allows us to faithfully capture distinct components of the underlying data and also flexibly adapt to new data by readily fine-tuning the neural basis functions. Attributed to the elaborately designed block terms, the suggested NeuApprox enjoys strong approximation ability and flexible data adaptation ability over the hand-crafted basis function-based methods. We also theoretically prove that NeuApprox can approximate any multivariate continuous function to arbitrary accuracy. Extensive experiments on diverse multi-dimensional datasets (including multispectral images, light field data, videos, traffic data, and point cloud data) demonstrate the promising performance of NeuApprox in terms of both approximation capability and adaptability.

تم تقديم نموذج جديد لتقريب الدوال متعددة المتغيرات، يعرف باسم التقريب العصبي (NeuApprox)، الذي يستخدم الشبكات العصبية غير المدربة كوظائف أساسية لتحسين تكيف البيانات وقدرات التقريب. هذه الطريقة تقوم بتفكيك الدوال المعقدة إلى مصطلحات كتلية قابلة للتفسير، مما يسمح بتمثيل أفضل للهياكل الأساسية للبيانات.

Se ha introducido un nuevo paradigma para la aproximación de funciones multivariadas, conocido como aproximación neuronal (NeuApprox), que utiliza redes neuronales no entrenadas como funciones base para mejorar la adaptación de datos y las capacidades de aproximación. Este enfoque descompone funciones complejas en términos de bloques interpretables, permitiendo una mejor representación de las estructuras de datos subyacentes.

Un nouveau paradigme pour l'approximation de fonctions multivariées, connu sous le nom d'approximation neuronale (NeuApprox), a été introduit, utilisant des réseaux neuronaux non entraînés comme fonctions de base pour améliorer l'adaptation des données et les capacités d'approximation. Cette approche décompose des fonctions complexes en termes de blocs interprétables, permettant une meilleure représentation des structures de données sous-jacentes.

A new paradigm for multivariate function approximation, known as neural approximation (NeuApprox), has been introduced, leveraging untrained neural networks as basis functions to enhance data adaptation and approximation capabilities. This approach decomposes complex functions into interpretable block terms, allowing for better representation of underlying data structures.