Analysis of Positional Encoding Methods and Their Mathematical Principles in Large Language Models

Abstract

: Transformer architecture is an architectural phenomenon in natural language processing. The self-attention mechanism inherent to it does not consider the word order. Thus, positional encoding (PE) is required to provide the word vectors with sequential information. This essay is a review of the processing involved in positional encoding. It starts by presenting straightforward methods of absolute sinuoidal positional encoding and proceeds on to the relative positional encoding methods that are more recent, including Rotational Positional Embedding (RoPE), Attention with Linear Biases (ALiBi). In the paper, the author discusses the underlying mathematical concepts of these techniques, such as, periodic, complex rotation and linear distance penalties. In addition, this paper presents the recent methods of context length extension including YaRN and positional interpolation. The approaches assist in the longer sequences. Nonetheless, existing techniques are not free of several challenges, including stability when working with very long sequences, information consistency across layers, and effective use of computational resources. Finally, the paper suggests the research directions in future with the use of the fields like the operator theory and the differential geometry to develop a better, more integrated, and more cross-modal positional encoding method.