Inexact calculus of variations on the hyperspherical tangent bundle and its connections to the attention mechanism

We offer a theoretical mathematical background through Lagrangian optimization on the unit hyperspherical manifold and its tangential collection with application to the Transformer and its token space. Our methods are catered to the attention mechanism in a theoretical setting, but largely appeal to a broader mathematical lens as well. The Transformer, as a flow map, exists in the tangent fiber for each token along the high-dimensional unit sphere. The circumstance of the hypersphere across the latent data is reasonable due to the trained diagonal matrix equal to the identity, which has various empirical justifications. Thus, under the continuum limit of the dynamics, the latent vectors flow among the tangent bundle. Using these facts, we devise a mathematical framework focusing on the attention mechanism through calculus of variations. We develop a functional and show that the continuous flow map induced by the Transformer satisfies this functional, therefore attention can be viewed as a natural solver of a calculus of variations problem. We invent new scenarios of when our methods are applicable based on loss optimization with respect to path optimality. We derive the projected Euler-Lagrange equation under the specific flow map. The variant of the Euler-Lagrange equation we present has various appearances in literature, but, to our understanding, oftentimes not foundationally proven or under other specialized cases. Our overarching proof is new: our techniques are classical and the use of the flow map object is original. We provide several other relevant results, primarily ones specific to neural scenarios. In particular, much of our analysis will be attempting to quantify Transformer data in variational contexts under neural approximations.