Optimization and Sampling Under Symmetry: Part 1 -- Examples
summary
Starts from two concrete claims: minimizing vAv over the complex unit sphere equals the matrix's smallest eigenvalue by recasting rank‑one projections as a unitary orbit and using spectral decomposition plus the Birkhoff–von Neumann theorem, and the integral of exp(−vAv) over the complex sphere collapses to an explicit finite formula (HCIZ) — a sum/determinant in the eigenvalues. Proofs use reduces nonconvex orbits to polytopes, uses Birkhoff von Neumann and Laplace transforms, and Vandermonde/determinant identities. Results work cleanly for complex/unitary groups; real/orthogonal cases remain harder.
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chapters (10)
- 0:00Introduction and overview
- 1:18Defining symmetries
- 6:01Symmetries in algorithms
- 9:04Origins and mathematical framework
- 15:05Optimization and sampling examples
- 24:20Proof of the first theorem
- 35:46Generalizations and connections
- 44:26Proof of the second theorem
- 52:03Applications and deep structure
- 1:00:01Conclusion and Q&A
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