Optimization and Sampling Under Symmetry: Part 2 -- Lie Theory
summary
Lie theory turns the concrete claims into tools: minimizing a linear functional over a unitary conjugation orbit becomes adjoint orbit → convex polytope via a Cartan subalgebra and Weyl‑group convex hull (Schur–Horn/Cosine convexity), so eigenvalue minimization reduces to linear optimization over vertices. Sampling uses the Killing form and Haar measure to get a Harish‑Chandra/HCIZ finite‑sum formula for partition functions; practical sampling proceeds by the Rayleigh (Gelfand‑Tsetlin) map that pushes Haar to Lebesgue on a polytope. Results require compact groups; noncompact cases and a missing piecewise‑polynomial correction remain open.
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chapters (17)
- 0:00Introduction
- 0:36Optimization and Sampling
- 2:37Symmetry
- 5:43Outline
- 6:49unitary group
- 8:17tangent space
- 12:00Conjugation
- 16:39Inner Product
- 18:30Lie Theory
- 20:20Conjugation Action
- 24:20Lead Algebra
- 28:25Optimization Problem
- 40:25Optimization Examples
- 44:41Sampling Examples
- 47:37Invariant Measures
- 50:50General Horizontal Formula
- 54:46Open Problem
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