One Diagram Maps the Gauge Freedom of Legendre Transforms from Convex Analysis to Information Geometry
FrnkNlsn · x · 2026-10-15
A compact nested-set diagram organizes the hierarchy behind the gauge freedom of Legendre transforms: extended real-valued functions (F≤F as the best convex lower bound), proper l.s.c. convex functions Γ₀ (biconjugation becomes an involution), Legendre-type Γ₁ (Fenchel-Young/Bregman divergences, Clairaut-Schwarz-Young theorem), twice differentiable Γ₂ (information Hessian geometry, Levi-Civita connection, Hessian flat divergences), and thrice differentiable Γ₃ (cubic tensor, α-Hessian geometry families). A useful reference for convex duality and information geometry.
More from Research
- RExBench: best coding agent completes only 33% of autonomous research extensions — najoungkim · 2026-10-16
- Google's CLASHarness benchmarks multi-agent conflicts in computer use, finds frontier VLMs struggle — google · 2026-10-16
- Tsinghua team proves vanilla flow-matching sampling can't exceed constant-depth expressive power — SQZ · 2026-10-16
- APEX-Harm benchmark: script-based injection attacks easily compromise agents in shared workspaces — KAUST · 2026-10-16
- Residual-stream forecasts predict agent violations before they happen, USC study shows — USC · 2026-10-16
- KAUST's NoteBank adds sparse episodic memory to linear attention, boosting long-context for free — kaust-generative-ai · 2026-10-16