The concept of Numerical Span Dimension (NSdim) provides a critical framework for understanding the limitations of neural networks in terms of their approximation capacity. By recognizing that the NSdim is inherently bounded within a finite Nonlinear Parameter Space (NPspace), neural network designers can make informed decisions about architecture choices.
Architecture Optimization: Designers can optimize the width and depth of neural networks based on the NSdim. Since increasing the width beyond a certain point leads to redundancy and does not enhance expressive power, it is more effective to focus on a balanced architecture that maximizes the NSdim without unnecessary complexity. This can lead to more efficient networks that require fewer parameters, reducing computational costs and improving training times.
Regularization Techniques: Understanding the relationship between NSdim and NPspace can inform the application of regularization methods such as L1 and L2. By strategically reducing the NPspace, designers can enhance the generalization capabilities of the network while maintaining a sufficient approximation capacity. This insight can guide the selection of regularization parameters during training.
Layer Design: The findings suggest that depth has a more significant impact on approximation capacity than width. Therefore, incorporating more hidden layers with fewer neurons may yield better performance than simply increasing the number of neurons in a single layer. This approach can help maintain a high NSdim while avoiding the pitfalls of overfitting.
Activation Functions: The choice of activation functions can also be influenced by NSdim insights. Using analytic activation functions that contribute positively to the NSdim can enhance the network's ability to approximate complex functions effectively.
By leveraging these theoretical insights, neural network architectures can be designed to be both efficient and effective, maximizing their practical utility in various applications.