CAT0space relaxation and block triangularization of partitioned matrix 1 10th JapaneseHungarian Symposium on Discrete Mathematics and Its Applications May 23 2017 Budapest ID: 1045514
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1. Maximum vanishing subspace problem, CAT(0)-space relaxation, and block-triangularization of partitioned matrix110th Japanese-Hungarian Symposium on Discrete Mathematics and Its ApplicationsMay 23, 2017, Budapest, HungaryHiroshi HiraiUniversity of Tokyohirai@mist.i.u-tokyo.ac.jp Joint work with Masaki Hamada
2. DM-decomposition of matrixA canonical form under row/column permutationMatching/stable set problem in bipartite graph permutation matrix (Dulmage-Mendelsohn 1958)2
3. bipartite graph zero submatrixstable set 3The family of maximum stable sets distributive latticeA maximal chain DM-decompositionA polynomial time algorithm via bipartite matching (max.)(max.)********
4. DM-decomposition of partitioned matrix(Ito-Iwata-Murota 1994) matrix permutation nonsingular 4
5. row/columnpermutation5Example: (2,2,2;2,2,2) matrix over GF(2)
6. Maximum Vanishing Subspace Problemwhere (H.2016, Implicit in Ito-Iwata-Murota 1994)Max. s.t. subsp.subsp.6Rem: ---> bipartite stable set prob.
7. Vanishing subspace The family of max. vanishing subspace modular latticeA maximal chain DM-decomposition (max.)(max.)7row/col operation “within blocks”+ row/col permutation
8. nonsingular ---> eigenvalue comp.sizes of submatrices & base field are fixed (H-Nakashima 2017) 8Polytime algorithm for DM-decom. is not known in general Special cases:one submatrix ---> Gaussian eliminationeach submatrix is 1x1 ---> original DM-decom.each submatrix is *x1 ---> CCF of mixed matrixeach submatrix is rank-1 (H. 16)Murota-Iri-Nakamura 87matroid intersectionVCSP
9. Why are MVSP & DM-decomposition interesting?9Numerical computation vs. combinatorial optimizationSubmodular optimization on (infinite) modular lattice Finite case: Kuivinen 2009,2011, Fujishige et al. 2015, Suggest “vector-space version” of combinatorial optimization ?? , finite setsubsetcardinalityfinite-dimensionalvector space subspace dimension
10. 10Result (answering the problem of Ito-Iwata-Murota 1994)Q1. Is MVSP solvable in polynomial time ?Q2. Is DM-decom. obtained in polynomial time ?YES (Hamada-H. 2017)Still we do not know the complete answer but... DM-decom. eigenvalue problem, more difficult numerical problemA reasonable coarse decomposition --- quasi DM-decomposition --- is obtained in polytime by solving weighted-MVSP. --- generalizes original-DM & CCF
11. 11In the rest of the talk, we describe the outline of the proof Theorem (Hamada-H. 2017)MVSP is solvable in polynomial timeMax. s.t. subsp.subsp.Difficulty:Duality & LP/convex relaxation are not known polynomial number of arithmetic operations in
12. 12 s.t. w.r.t. reverse inclusion: modular lattice of all vector subsp. in w.r.t. inclusionLemma (cf. Iwata-Murota 1995) is submodular on MVSP is submodular optimization
13. Outline : Beyond Euclidean convex relaxation13Splitting Proximal Point Algorithm (Bačák, Ohta-Pálfia) to minimize sum of convex function in CAT(0)-space.Apply SPPA to CAT(0)-space relaxation of MVSP.Submodular optimization on Convex optimization on distributive latticeEuclidean spacemodular latticeCAT(0)-spaceLovász extension MVSP is submodular optimization on modular lattice.
14. 14CAT(0)-space (Gromov 87)Cartan-Alexandrov-TopogonovFACT: CAT(0)-space is uniquely geodesic ---> convex functioncurvature geodesic metric space in which every triangle is “slimmer”
15. Modular lattice CAT(0)-space 15: modular lattice of finite rank := the set of convex combinations of s.t. is a chain Theorem (Haettel-Kielak-Schwer 2017, Chalopin-Chepoi-H-Osajda 2014) is a complete CAT(0)-space. othoscheme complex(Brady-McCammond 2012)
16. Example16... ....
17. Submodular function convex function 17: modular lattice of finite rank Lovasz extension of Theorem (H. 2016) is submodular on is convex on (w.r.t. CAT(0)-metric) The original version:
18. MVSP convex optimization on CAT(0)-space 18 s.t. We apply continuous optimization algorithm to this problem
19. 19Lemma , If some is a minimizer of To recover a minimizer of f from , we use:
20. Theorem (Ohta-Pálfia 15)-Lipschitz, -strongly-convex, Splitting Proximal Point Algorithm20 complete CAT(0)-space ,..., convex functions on Goal: minimize SPPA: (Bačák 14)
21. 21Apply SPPA to perturbed objective ensure-strong-convexity each term is -Lipschitz generated by SPPA for minimizer
22. 22In each iteration of SPPA, we need to solve:P1: Min. P2: Min. s.t. rev. inclusion: modular lattice of all vector subsp. in : bilinear form P1 and P2are solvable inpolynomial time
23. 23P1: Min. Minimizer exists in the simplex ---> easy convex quadratic program
24. 24P2: Min. s.t. : orthogonal space w.r.t. Minimizer exists in ---> easy convex quadratic program distributive lattice
25. 25Concluding remarksWeighted-MVSP: Max. is solvable in pseudo-polynomial timeQuasi DM-decomposition a chain of max. vanishing subsp. detectable by WMVSPDuality theory + better algorithm for MVSP ? Vector-space generalization of other problems ? CAT(0)-space relaxation ---> new paradigm in combinatorial optimization ?
26. 26Thank you for your attention!M. Hamada and H. Hirai: Maximum vanishing subspace problem, CAT(0)-space relaxation, and block-triangularization of partitioned matrix, 2017, arXiv.