Transformer Alignment in Large Language Models

Large Language Models (LLMs) have made significant strides in natural language processing, and a precise understanding of the internal mechanisms driving their success is essential. We regard LLMs as transforming embeddings via a discrete, coupled, nonlinear, dynamical system in high dimensions. This perspective motivates tracing the trajectories of individual tokens as they pass through transformer blocks, and linearizing the system along these trajectories through their Jacobian matrices. In our analysis of 38 openly available LLMs, we uncover the alignment of top left and right singular vectors of Residual Jacobians, as well as the emergence of linearity and layer-wise exponential growth. Notably, we discover that increased alignment $\textit{positively correlates}$ with model performance. Metrics evaluated post-training show significant improvement in comparison to measurements made with randomly initialized weights, highlighting the significant effects of training in transformers. These findings reveal a remarkable level of regularity that has previously been overlooked, reinforcing the dynamical interpretation and paving the way for deeper understanding and optimization of LLM architectures.