Sampling Matrices From Harish-Chandra-Itzykson-Zuber Densities

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Proves two efficient samplers for HCIZ/exponential measures on unitary orbits: one achieves total-variation error C in time polynomial in log(1/C), the other achieves infinity-divergence error C in time polynomial in 1/C. Key tools are projecting the orbit to the Gelfand–Tsetlin polytope (density becomes log‑concave), sampling there via hit-and-run or a grid walk, then reconstructing the matrix by sampling a product of spheres fiber one principal minor at a time via characteristic‑polynomial formulas. Works for DP low‑rank approximation and max‑entropy quantum decompositions; improving runtimes and orthogonal cases remain open.

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1. 0:00Intro
2. 0:56Background: Differential Privacy
3. 4:43Motivation: Differential Privacy
4. 7:28Motivation: Quantum Inference
5. 10:27Sampling from Unitary Orbits
6. 13:32Main Result of This Talk
7. 16:07HCIZ Is Everywhere
8. 21:02Searching for Self-Reducibility
9. 26:08A Spectral Reference Frame
10. 30:35Rank One Case
11. 32:59Rayleigh Map and GT Polytopes Rayleigh Mop
12. 38:05Rayleigh Map and GT Polytopes Rayleigh Map
13. 40:08Step 2: From Polytope to Orbit
14. 44:25Conclusion