Applied Hodge Theory

on YouTube ↗43m

" summary

Linear-algebra Hodge theory on clique complexes decomposes pairwise edge data exactly into orthogonal components: a solvable gradient (global) component found by solving a graph Laplacian (nearly-linear solvers), a curl (local inconsistency) component supported on triangles, and a harmonic (circular) component whose nonzero dimension equals loop count and produces cyclic rankings. Persistent homology tracks when loops appear (polynomial-time), upper Laplacians can be non–diagonally dominant and harder to solve, and finite games map to a unique potential+harmonic+neutral split, making correlated equilibria LP-tractable while Nash remains hard.

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" chapters (28)

1. 0:00Intro
2. 1:07Helmholtz-Hodge Decomposition
3. 2:11Hodge Theory in Linear Algebra
4. 8:11Classical Hodge Theory on Riemannian Manifolds
5. 8:53Hodge Theory on Metric Spaces
6. 9:39Combinatorial Hodge Theory on Simplicial Complexes
7. 10:44Example: graph and clique complex
8. 12:29Hodge Laplacian
9. 13:48Applications of Hodge Decomposition
10. 14:15Optical Flow Decomposition and Regularization Rudin Osher Fatemi 1992 piecewise constant flows
11. 15:59Example: periodic motions are harmonic
12. 16:32Adelson's illusion in Computer Vision
13. 18:14Retinex Theorey based on Approximation of Gradient Flows
14. 18:37Crowdsourcing QoE evaluation of Multimedia
15. 20:19Paired comparison data on graphs
16. 22:34Hodge Decomposition on Graphs Jiang-Lim-Y-Ye'11
17. 25:37Computation Issue of Curl and Harmonics
18. 27:33Topological Obstructions
19. 28:45Basic Problems in Hodge Rank
20. 29:37Strategic Simplicial Complex for Games
21. 32:27Nash and Correlated Equilibrium
22. 33:33Why Correlated Equilibria?
23. 34:45Low Rank CE Tensors
24. 36:24Another simplification: Graphical Games
25. 37:30Hodge Decomposition of Finite Games
26. 38:59Bimatrix Games
27. 40:20Summary
28. 41:37Acknowledgement

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