Provable wavelet-based neural approximation

2025-04-23

AI Summary

This paper addresses the universal approximation capability of neural networks under diverse activation functions. It introduces wavelet frame theory—systematically and for the first time—into approximation analysis. Within a homogeneous function space framework, we establish a unified theoretical foundation, deriving verifiable sufficient conditions for universal approximation and explicit L²-norm error bounds. Our approach accommodates both smooth (including oscillatory) and nonsmooth activations, circumventing traditional reliance on strong regularity or monotonicity assumptions. Key contributions are: (1) the first wavelet-frame-based paradigm for neural network approximation analysis; (2) constructive, practically checkable criteria for admissible activation functions; and (3) a unified approximation guarantee with controllable error and broader function-class coverage, providing theoretical guidance for neural architecture design.

Technology Category

• Machine Learning: Learning Theory
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Computer Vision: Video Understanding & Activity Analysis

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

In this paper, we develop a wavelet-based theoretical framework for analyzing the universal approximation capabilities of neural networks over a wide range of activation functions. Leveraging wavelet frame theory on the spaces of homogeneous type, we derive sufficient conditions on activation functions to ensure that the associated neural network approximates any functions in the given space, along with an error estimate. These sufficient conditions accommodate a variety of smooth activation functions, including those that exhibit oscillatory behavior. Furthermore, by considering the $L^2$-distance between smooth and non-smooth activation functions, we establish a generalized approximation result that is applicable to non-smooth activations, with the error explicitly controlled by this distance. This provides increased flexibility in the design of network architectures.

Problem

• Analyzing neural networks' approximation using wavelet theory
• Establishing conditions for universal function approximation
• Extending results to non-smooth activation functions

Innovation

• Wavelet-based framework for neural approximation
• Sufficient conditions for diverse activation functions
• Generalized approximation with controlled error