Analogies between Transformer Layers and Power Method
This provides a theoretical understanding of transformer dynamics for ML researchers, but the results are primarily conceptual and lack concrete performance improvements.
The paper reveals an analogy between transformer layers and the power method, showing that tokens align with the principal eigenvector of the product of output and value weight matrices. In shared-weight transformers, this alignment is empirically and analytically evident, and the analogy enables steering transformer output toward arbitrary directions.
In the paper we show that there is an analogy between the operations occurring in a layer of a transformer (projections and layer normalizations, disregarding the feedforward neural network) and a step in the power method. Coherently with this analogy, we show that passing through a layer the tokens tend to be tilted towards the principal eigenvector of a matrix which is the product of the output and value weight matrices of that layer. In the special case of a transformer with shared weights (i.e., in which all layers have identical weights) then the alignment with this principal eigenvector is particularly evident empirically, and can also be shown analytically. The analogy also suggests a method to steer the output of the transformer towards an arbitrary desired direction in token space.