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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.11573 |
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| _version_ | 1866911266320154624 |
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| author | Lee-Jenkins, Christopher R. |
| author_facet | Lee-Jenkins, Christopher R. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_11573 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Softmax as a Lagrangian-Legendrian Seam Lee-Jenkins, Christopher R. Machine Learning 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. |
| title | Softmax as a Lagrangian-Legendrian Seam |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2511.11573 |