{"ID":2869641,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2511.11573","arxiv_id":"2511.11573","title":"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.","short_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\" o...","url_abs":"https://arxiv.org/abs/2511.11573","url_pdf":"https://arxiv.org/pdf/2511.11573v1","authors":"[\"Christopher R. Lee-Jenkins\"]","published":"2025-09-17T02:42:28Z","proceeding":"cs.LG","tasks":"[\"cs.LG\"]","methods":"[]","has_code":false}