This paper provides an in-depth analysis of Wave Network, a novel token representation method derived from the Wave Network, designed to capture both global and local semantics of input text through wave-inspired complex vectors. In complex vector token representation, each token is represented with a magnitude component, capturing the global semantics of the entire input text, and a phase component, encoding the relationships between individual tokens and the global semantics. Building on prior research that demonstrated the effectiveness of wave-like operations, such as interference and modulation, during forward propagation, this study investigates the convergence behavior, backpropagation characteristics, and embedding independence within the Token2Wave framework. A detailed computational complexity analysis shows that Token2Wave can significantly reduce video memory usage and training time compared to BERT. Gradient comparisons for the [CLS] token, total input text, and classifier parameters further highlight Token2Wave's unique characteristics. This research offers new insights into wave-based token representations, demonstrating their potential to enable efficient and computationally friendly language model architectures.

We propose an innovative token representation and update method in a new ultra-small language model: the Wave network. Specifically, we use a complex vector to represent each token, encoding both global and local semantics of the input text. A complex vector consists of two components: a magnitude vector representing the global semantics of the input text, and a phase vector capturing the relationships between individual tokens and global semantics. Experiments on the AG News text classification task demonstrate that, when generating complex vectors from randomly initialized token embeddings, our single-layer Wave Network achieves 90.91% accuracy with wave interference and 91.66% with wave modulation - outperforming a single Transformer layer using BERT pre-trained embeddings by 19.23% and 19.98%, respectively, and approaching the accuracy of the pre-trained and fine-tuned BERT base model (94.64%). Additionally, compared to BERT base, the Wave Network reduces video memory usage and training time by 77.34% and 85.62% during wave modulation. In summary, we used a 2.4-million-parameter small language model to achieve accuracy comparable to a 100-million-parameter BERT model in text classification.

Graph algorithms are key tools in many fields of science and technology. Some of these algorithms depend on propagating information between distant nodes in a graph. Recently, there have been a number of deep learning architectures proposed to learn on undirected graphs. However, most of these architectures aggregate information in the local neighborhood of a node, and therefore they may not be capable of efficiently propagating long-range information. To solve this problem we examine a recently proposed architecture, wave, which propagates information back and forth across an undirected graph in waves of nonlinear computation. We compare wave to graph convolution, an architecture based on local aggregation, and find that wave learns three different graph-based tasks with greater efficiency and accuracy. These three tasks include (1) labeling a path connecting two nodes in a graph, (2) solving a maze presented as an image, and (3) computing voltages in a circuit. These tasks range from trivial to very difficult, but wave can extrapolate from small training examples to much larger testing examples. These results show that wave may be able to efficiently solve a wide range of problems that require long-range information propagation across undirected graphs. An implementation of the wave network, and example code for the maze problem are included in the tflon deep learning toolkit (https://bitbucket.org/mkmatlock/tflon).