Softmax as a Lagrangian-Legendrian Seam
Abstract
This note offers a first bridge from machine learning to modern differential geometry. We show that the logits-to-probabilities step implemented by softmax can be modeled as a geometric interface: two potential-generated, conservative descriptions (from negative entropy and log-sum-exp) meet along a Legendrian "seam" on a contact screen (the probability simplex) inside a simple folded symplectic collar. Bias-shift invariance appears as Reeb flow on the screen, and the Fenchel-Young equality/KL gap provides a computable distance to the seam. We work out the two- and three-class cases to make the picture concrete and outline next steps for ML: compact logit models (projective or spherical), global invariants, and connections to information geometry where on-screen dynamics manifest as replicator flows.
Cite
@article{arxiv.2511.11573,
title = {Softmax as a Lagrangian-Legendrian Seam},
author = {Christopher R. Lee-Jenkins},
journal= {arXiv preprint arXiv:2511.11573},
year = {2025}
}