# Inexact calculus of variations on the hyperspherical tangent bundle with connections to the attention mechanism

arXiv: https://arxiv.org/abs/2507.15431 · HTML: https://mlsanity.com/paper/2507.15431/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 26 references that are in the map.
- First posted on arXiv 21 Jul 2025.

## Builds on (most cited first, 4 of 4)

- [Plug and Play Language Models: A Simple Approach to Controlled Text Generation](https://mlsanity.com/paper/1912.02164/) (2019)
- [A mathematical perspective on Transformers](https://mlsanity.com/paper/2312.10794/) (2023)
- [The Affine Geometric Heat Flow and Motion Planning for Dynamic Systems](https://mlsanity.com/paper/1905.02321/) (2019)
- [Existence of optimal pairs for optimal control problems with states constrained to Riemannian manifolds](https://mlsanity.com/paper/2401.07006/) (2024)

## Abstract (arXiv)

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.