Simple initialization and parametrization of sinusoidal networks via their kernel bandwidth

TL;DR

This paper introduces a simplified sinusoidal neural network model, analyzes its kernel behavior, and uses these insights to improve initialization, enhancing performance in learning implicit models and differential equations.

Contribution

It proposes a simplified sinusoidal network, analyzes its kernel as a low-pass filter, and develops an improved initialization method based on kernel bandwidth insights.

Findings

Kernel approximates a low-pass filter with adjustable bandwidth

Simplified network facilitates practical implementation and theoretical analysis

Optimized initialization improves performance on implicit models and differential equations

Abstract

Neural networks with sinusoidal activations have been proposed as an alternative to networks with traditional activation functions. Despite their promise, particularly for learning implicit models, their training behavior is not yet fully understood, leading to a number of empirical design choices that are not well justified. In this work, we first propose a simplified version of such sinusoidal neural networks, which allows both for easier practical implementation and simpler theoretical analysis. We then analyze the behavior of these networks from the neural tangent kernel perspective and demonstrate that their kernel approximates a low-pass filter with an adjustable bandwidth. Finally, we utilize these insights to inform the sinusoidal network initialization, optimizing their performance for each of a series of tasks, including learning implicit models and solving differential equations.