1. Maximum vanishing subspace problem, CAT(0)-space relaxation, and block-triangularization of partitioned matrix
Hiroshi Hirai
University of Tokyo
Joint work with Masaki Hamada
10th Japanese-Hungarian Symposium on
Discrete Mathematics and Its Applications
May 23, 2017, Budapest, Hungary

2. DM-decomposition of matrix
(Dulmage-Mendelsohn 1958)
A canonical form under row/column permutation
Matching/stable set problem in bipartite graph ∗ ∗ ∗
𝐴 ↦𝑃𝐴𝑄= ∗ ∗ ∗ ∗
𝑃,𝑄:permutation matrix ∗ ∗

3. 𝑋 𝑌 * * * * * * 𝑋 * * 𝑌 𝐴=( 𝑎 𝑖𝑗 ) bipartite graph zero submatrix
edge 𝑖𝑗 ⟺ 𝑎 𝑖𝑗 ≠0 𝑋 𝑌 1′
1 ′ 2 ′ 3 ′ 1 1 * 2′ 2 * * 2
𝑋 0
𝑋 1
𝑋 2
𝑋 3 3′ * * 3 * 𝑋 * * 𝑌
𝑌 0
𝑌 1
𝑌 2
𝑌 3
𝐴=( 𝑎 𝑖𝑗 )
bipartite graph
zero submatrix
stable set
𝑋,𝑌 , 𝑋 ′ , 𝑌 ′ :stable⇒ 𝑋∪ 𝑋 ′ ,𝑌∩ 𝑌 ′ , 𝑋∩ 𝑋 ′ ,𝑌∪ 𝑌 ′ :stable
The family of maximum stable sets ⇒ distributive lattice
A maximal chain ⇒ DM-decomposition
A polynomial time algorithm via bipartite matching
(max.)

4. DM-decomposition of partitioned matrix
(Ito-Iwata-Murota 1994)
𝐴= 𝐴 11 𝐴 12 𝐴 21 𝐴 22 ⋯ 𝐴 1𝜈 𝐴 2𝜈 ⋮ ⋱ ⋮ 𝐴 𝜇1 𝐴 𝜇2 ⋯ 𝐴 𝜇𝜈
𝐴 𝛼𝛽 : 𝑚 𝛼 × 𝑛 𝛽 matrix
↦𝑃 𝐸 𝐸 2 ⋯ ⋮ ⋱ ⋮ ⋯ 𝐸 𝜇 𝐴 11 𝐴 12 𝐴 21 𝐴 22 ⋯ 𝐴 1𝜈 𝐴 2𝜈 ⋮ ⋱ ⋮ 𝐴 𝜇1 𝐴 𝜇2 ⋯ 𝐴 𝜇𝜈 𝐹 𝐹 2 ⋯ ⋮ ⋱ ⋮ ⋯ 𝐹 𝜈 𝑄 ∗
𝑃,𝑄: permutation
𝐸 𝛼 , 𝐹 𝛽 :nonsingular =

5. Example: (2,2,2;2,2,2) matrix over GF(2) =
row/column
permutation

6. Maximum Vanishing Subspace Problem
(H.2016, Implicit in Ito-Iwata-Murota 1994)
𝐴= 𝐴 11 𝐴 12 𝐴 21 𝐴 22 ⋯ 𝐴 1𝜈 𝐴 2𝜈 ⋮ ⋱ ⋮ 𝐴 𝜇1 𝐴 𝜇2 ⋯ 𝐴 𝜇𝜈
𝐴 𝛼𝛽 : 𝑚 𝛼 × 𝑛 𝛽 matrix over field 𝔽
Max. 𝛼 dim 𝑋 𝛼 + 𝛽 dim 𝑌 𝛽
s.t.
𝑋 𝛼 ⊆ 𝔽 𝑚 𝛼 , 𝑌 𝛽 ⊆ 𝔽 𝑛 𝛽 ,
𝐴 𝛼𝛽 𝑋 𝛼 , 𝑌 𝛽 = ∀𝛼,𝛽 ,
subsp.
where 𝐴 𝛼𝛽 : 𝔽 𝑚 𝛼 × 𝔽 𝑛 𝛽 →𝔽
𝑢,𝑣 ↦ 𝑢 𝑇 𝐴 𝛼𝛽 𝑣
Rem: 𝑚 𝛼 = 𝑛 𝛽 =1 ---> bipartite stable set prob.

7. 𝐴 ~ Vanishing subspace (𝑋,𝑌)⇔ 𝑋 𝑌 𝑋= 𝑋 1 ⊕ 𝑋 2 ⊕⋯⊕ 𝑋 𝜇
𝑌= 𝑌 1 ⊕ 𝑌 2 ⊕⋯⊕ 𝑌 𝜈
𝐴 𝛼𝛽 𝑋 𝛼 , 𝑌 𝛽 = ∀𝛼,𝛽
row/col operation “within blocks”
+ row/col permutation
𝐴 ~
← dim 𝑋→ 𝑋
⟵ dim 𝑌 ⟶ 𝑌
𝑋,𝑌 , 𝑋 ′ , 𝑌 ′ :vanishing
⇒ 𝑋+ 𝑋 ′ ,𝑌∩ 𝑌 ′ , 𝑋∩ 𝑋 ′ ,𝑌+ 𝑌 ′ :vanishing
The family of max. vanishing subspace ⇒ modular lattice
A maximal chain ⇒ DM-decomposition
(max.)

8. Polytime algorithm for DM-decom. is not known in general
Special cases:
one submatrix > Gaussian elimination
each submatrix is 1x > original DM-decom.
each submatrix is *x > CCF of mixed matrix
each submatrix is rank-1 (H. 16)
Murota-Iri-Nakamura 87
matroid
intersection
𝐴 11 𝐴 12 𝐴 21 𝐴 22 , 𝐴 𝛼𝛽 : nonsingular ---> eigenvalue comp.
sizes of submatrices & base field are fixed
(H-Nakashima 2017)
VCSP

9. Why are MVSP & DM-decomposition interesting?
Numerical computation vs. combinatorial optimization
Submodular optimization on (infinite) modular lattice
　 Finite case: Kuivinen 2009,2011, Fujishige et al. 2015, ( ) 𝑛
Suggest “vector-space version”
of combinatorial optimization ?? ⋯
𝑆 ⊆𝑉, |𝑆|
subset
finite set
cardinality
finite-dimensional
vector space
subspace
dim S
dimension

10. Result (answering the problem of Ito-Iwata-Murota 1994)
Q1. Is MVSP solvable in polynomial time ?
YES (Hamada-H. 2017)
Q2. Is DM-decom. obtained in polynomial time ?
Still we do not know the complete answer but...
DM-decom. ⊇ eigenvalue problem,
more difficult numerical problem
A reasonable coarse decomposition
--- quasi DM-decomposition ---
is obtained in polytime by solving weighted-MVSP.
--- generalizes original-DM & CCF

11. MVSP is solvable in polynomial time
In the rest of the talk, we describe the outline of the proof
Theorem (Hamada-H. 2017)
MVSP is solvable in polynomial time
⊇ polynomial number
of arithmetic operations in 𝔽
Max. 𝛼 dim 𝑋 𝛼 + 𝛽 dim 𝑌 𝛽
s.t.
𝑋 𝛼 ⊆ 𝔽 𝑚 𝛼 , 𝑌 𝛽 ⊆ 𝔽 𝑛 𝛽
𝐴 𝛼𝛽 𝑋 𝛼 , 𝑌 𝛽 = ∀𝛼,𝛽
subsp.
Difficulty:
Duality & LP/convex relaxation are not known

12. MVSP is submodular optimization
𝑚+𝑛+1
Min. − 𝛼 dim 𝑋 𝛼 − 𝛽 dim 𝑌 𝛽 +𝑀 𝛼,𝛽 rank 𝐴 𝛼𝛽 | 𝑋 𝛼 × 𝑌 𝛽
s.t.
( 𝑋 1 , 𝑋 2 ,…, 𝑋 𝜇 , 𝑌 1 , 𝑌 2 ,…, 𝑌 𝜈 )
∈ ℒ 1 × ℒ 2 ×…× ℒ 𝜇 × ℳ 1 × ℳ 2 ×…× ℳ 𝜈
ℒ 𝛼 / ℳ 𝛽 : modular lattice of all vector subsp. in 𝔽 𝑚 𝛼 / 𝔽 𝑛 𝛽
w.r.t.
inclusion
w.r.t. reverse
inclusion
𝑚≔ 𝛼 𝑚 𝛼 ,𝑛≔ 𝛽 𝑛 𝛽
Lemma (cf. Iwata-Murota 1995)
𝑋 𝛼 × 𝑌 𝛽 ⟼rank 𝐴 𝛼𝛽 | 𝑋 𝛼 × 𝑌 𝛽 is submodular on ℒ 𝛼 × ℳ 𝛽

13. Outline : Beyond Euclidean convex relaxation
Submodular optimization on
distributive lattice
modular lattice ↪
Lovász extension ↪
Convex optimization on
Euclidean space
CAT(0)-space
MVSP is submodular optimization on modular lattice.
Splitting 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.

14. CAT(0)-space (Gromov 87) ⇔ geodesic metric space (𝑆,𝑑) in which
Cartan-Alexandrov-Topogonov
curvature ≤0
CAT(0)-space (Gromov 87)
⇔ geodesic metric space (𝑆,𝑑) in which
every triangle is “slimmer” 𝑥 𝑦 𝑧
ℝ 2
𝑑 𝑥,𝑦 = 𝑥 − 𝑦 2
𝑑 𝑦,𝑧 = 𝑦 − 𝑧 2
𝑑 𝑧,𝑥 = 𝑧 − 𝑥 2 𝑝 𝑥 𝑆 𝑝 𝑦 𝑧
𝑑 𝑥,𝑝 ≤ 𝑥 − 𝑝 2
FACT: CAT(0)-space is uniquely geodesic
---> convex function

15. Modular lattice ↪ CAT(0)-space
othoscheme complex
ℒ: modular lattice of finite rank
𝐾 ℒ := the set of convex combinations 𝑝∈ℒ 𝜆 𝑝 𝑝 of ℒ
s.t. supp λ is a chain
(Brady-McCammond 2012)
𝑝 3
(ℝ 𝑛 , 𝑙 2 )
000
100
110
111
𝑝 2 ℒ
𝑝 1
𝑝 0
Theorem (Haettel-Kielak-Schwer 2017, Chalopin-Chepoi-H-Osajda 2014)
𝐾 ℒ is a complete CAT(0)-space.

16. Example
....
... ℒ
𝐾 ℒ
ℒ= 2 {1,2,…,𝑛}
𝐾 ℒ ≃ 0,1 𝑛

17. Submodular function ↪ convex function
ℒ: modular lattice of finite rank
Lovasz extension 𝑓 :𝐾(ℒ)→ℝ of 𝑓:ℒ→ℝ
𝑓 𝑥 ≔ 𝑖 𝜆 𝑖 𝑓( 𝑝 𝑖 ) (𝑥= 𝑖 𝜆 𝑖 𝑝 𝑖 ∈𝐾(ℒ))
Theorem (H. 2016)
𝑓 is submodular on ℒ
⇔ 𝑓 is convex on 𝐾(ℒ) (w.r.t. CAT(0)-metric)
The original version: ℒ= 2 1,2,…,𝑛 ,𝐾 ℒ ≃ [0,1] 𝑛

18. MVSP ↪ convex optimization on CAT(0)-space
Min. − 𝛼 dim ( 𝑥 𝛼 )− 𝛽 dim ( 𝑦 𝛽 ) +𝑀 𝛼,𝛽 𝑅 𝛼𝛽 ( 𝑥 𝛼 , 𝑦 𝛽 )
s.t.
( 𝑥 1 , 𝑥 2 ,…, 𝑥 𝜇 , 𝑦 1 , 𝑦 2 ,…, 𝑦 𝜈 )
∈ 𝐾(ℒ 1 × ℒ 2 ×…× ℒ 𝜇 × ℳ 1 × ℳ 2 ×…× ℳ 𝜈 )
𝐾(ℒ 1 )×𝐾( ℒ 2 )×…× 𝐾(ℒ 𝜇 )× 𝐾(ℳ 1 )× 𝐾(ℳ 2 )×…× 𝐾(ℳ 𝜈 ) ≅
𝑅 𝛼𝛽 𝑋 𝛼 , 𝑌 𝛽 ≔ rank 𝐴 𝛼𝛽 | 𝑋 𝛼 × 𝑌 𝛽
We apply continuous optimization algorithm
to this problem

19. To recover a minimizer of f from 𝑓 , we use:
Lemma 𝑓:ℒ→ℤ, 𝑥= 𝜆 𝑖 𝑝 𝑖 ∈𝐾 ℒ
If 𝑓 𝑥 −min 𝑓<1
⇒ some 𝑝 𝑖 is a minimizer of 𝑓

20. Splitting Proximal Point Algorithm
(Bačák 14)
𝑆,𝑑 : complete CAT(0)-space
𝑓 1 , 𝑓 2 ,..., 𝑓 𝑁 : convex functions on 𝑆
Goal: minimize 𝑓≔ 𝑖=1 𝑁 𝑓 𝑖
SPPA: 𝑧 𝑘+1 ≔ argmin 𝑧 ∈ 𝑆 𝑓 𝑘 mod 𝑁 𝑧 𝜆 𝑘 𝑑(𝑧, 𝑧 𝑘 ) 2
Theorem (Ohta-Pálfia 15)
𝑓 𝑖 :𝐿-Lipschitz, 𝑓:𝜖-strongly-convex, 𝜆 𝑘 ≔ 1 2𝜖 𝑘+1
⇒ 𝑓 𝑧 𝑘 −min 𝑓≤ poly(𝐿, 𝑁, 𝜖 −1 , diam 𝑆) 𝑘

21. Apply SPPA to perturbed objective 𝛼 − dim 𝑥 𝛼 +𝜖 𝑑(0, 𝑥 𝛼 ) 2
+ 𝛼,𝛽 𝑀 𝑅 𝛼𝛽 ( 𝑥 𝛼 , 𝑦 𝛽 )
ensure
𝜖-strong-convexity
𝜖 −1 =𝑂(𝑛+𝑚)
𝑧=( 𝑥 1 , 𝑥 2 ,…, 𝑥 𝜇 , 𝑦 1 , 𝑦 2 ,…, 𝑦 𝜈 )
0≤ obj 𝑧 −obj(𝑧)≤1/2
each term is 𝑂 poly 𝑚,𝑛 -Lipschitz
𝑁=𝑂(𝑚𝑛)
𝑧 𝑘 = generated by SPPA for obj
𝑘=Ω poly 𝑚,𝑛
⇒obj 𝑧 𝑘 −opt≤ obj 𝑧 𝑘 − opt <1
⇒supp( 𝑧 𝑘 )∋ minimizer

22. In each iteration of SPPA, we need to solve:
P1: Min. − dim 𝑥 +𝜖𝑑 0,𝑥 𝜆 𝑑 𝑥 0 ,𝑥 2
s.t. 𝑥 ∈ 𝐾 ℒ
P2: Min. 𝑅 𝐴 𝑥,𝑦 + 1 2𝜆 𝑑 𝑥 0 ,𝑥 2 +𝑑 𝑦 0 ,𝑦 2
s.t 𝑥,𝑦 ∈𝐾 ℒ×ℳ ≅𝐾 ℒ)×𝐾(ℳ
rev. inclusion
ℒ /ℳ: modular lattice of all vector subsp. in 𝔽 𝑚 / 𝔽 𝑛
𝐴: 𝔽 𝑚 × 𝔽 𝑛 →𝔽: bilinear form
𝑅 𝐴 𝑋,𝑌 ≔rank 𝐴 ​ 𝑋×𝑌
P1 and P2
are solvable in
polynomial time

23. Minimizer 𝑥 ∗ exists in the simplex ∋ 𝑥 0
1 2𝜆 𝑥 0 −𝑥 2 2
− 𝑥 1
𝜖 𝑥 2 2
P1: Min. − dim 𝑥 +𝜖𝑑 0,𝑥 𝜆 𝑑 𝑥 0 ,𝑥 2
s.t. 𝑥 ∈ 𝐾 ℒ
𝑝 3
𝑥 0 = 𝑖 𝜆 𝑖 𝑝 𝑖
𝑥 ∗
𝑝 2
𝑝 0
𝑝 1
Minimizer 𝑥 ∗ exists in the simplex ∋ 𝑥 0
---> easy convex quadratic program

24. ℒ ℳ P2: Min. 𝑅 𝐴 𝑥,𝑦 + 1 2𝜆 𝑑 𝑥 0 ,𝑥 2 +𝑑 𝑦 0 ,𝑦 2 s.t. 𝑥,𝑦 ∈𝐾 ℒ×ℳ
X 0⊥
X 0
Y 0⊥
Y 0
∙ ⊥ : orthogonal space w.r.t. 𝐴
distributive lattice
Minimizer ( 𝑥 ∗ , 𝑦 ∗ ) exists
in 𝐾 X 0 , Y 0⊥ ×𝐾 X 0⊥ , Y 0 ↪ 0,1 𝑚+𝑛
---> easy convex quadratic program

25. Concluding remarks Weighted-MVSP: Max. 𝛼 𝐶 𝛼 dim 𝑋 𝛼 + 𝛽 𝐷 𝛽 dim 𝑌 𝛽
is solvable in pseudo-polynomial time
Quasi DM-decomposition
⟺ a chain of max. vanishing subsp. detectable by WMVSP
Duality theory + better algorithm for MVSP ?
Vector-space generalization of other problems ?
CAT(0)-space relaxation ---> new paradigm
in combinatorial optimization ?

26. Thank 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.