Inexact calculus of variations on the hyperspherical tangent bundle with connections to the attention mechanism
Andrew Gracyk
Oct 2026
Machine Learning
Abstract
We offer a theoretical mathematical background through Lagrangian optimization on the unit hyperspherical manifold and its tangential structure. Our methods can be categorized as inexact since our methods are projection-based and since we will perturb the functional optimization with epsilon-type quantities. We draw connections to the attention mechanism and the Transformer since it exists as a flow map in the tangent fiber for each token along the high-dimensional unit sphere. Our motivation for this work is primarily twofold: we study the attention mechanism under its flow map and its relations to traditional calculus of variations and Lagrangian optimization; and we study several calculus problems of variations on the unit hypersphere that appeal to a broader mathematical lens in approximating, variational contexts.
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