Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized...
Transcript of Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized...
![Page 1: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/1.jpg)
Semidefinite Programming Relaxations for RecoveringHidden Communities
Jiaming Xu
Krannert School of ManagementPurdue University
Joint work with Bruce Hajek (Illinois) and Yihong Wu (Yale)
December 17, 2016
![Page 2: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/2.jpg)
Community detection in networks
• Observe local pairwise interactions between objects, e.g., socialnetworks, biological networks ...
• Interested in global properties of objects, e.g., similarity
Community detection in networks
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Community detection in networks
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Goal: identify communities of similar objects, related to clustering andgraph partitioning.
Jiaming Xu (Purdue) SDP for community detection 2
![Page 3: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/3.jpg)
Community detection in networks
• Observe local pairwise interactions between objects, e.g., socialnetworks, biological networks ...
• Interested in global properties of objects, e.g., similarity
Community detection in networks
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Community detection in networks
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Goal: identify communities of similar objects, related to clustering andgraph partitioning.
Jiaming Xu (Purdue) SDP for community detection 2
![Page 4: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/4.jpg)
Community detection in networks
• Observe local pairwise interactions between objects, e.g., socialnetworks, biological networks ...
• Interested in global properties of objects, e.g., similarity
Community detection in networks
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Community detection in networks
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Goal: identify communities of similar objects, related to clustering andgraph partitioning.
Jiaming Xu (Purdue) SDP for community detection 2
![Page 5: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/5.jpg)
Stochastic block model [Holland-Laskey-Leinhardt ’83]
Planted partition model [Condon-Karp 01’]
Jiaming Xu (Purdue) SDP for community detection 3
![Page 6: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/6.jpg)
Stochastic block model [Holland-Laskey-Leinhardt ’83]
Planted partition model [Condon-Karp 01’]
p = 0.8
q = 0.09
Jiaming Xu (Purdue) SDP for community detection 4
![Page 7: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/7.jpg)
Stochastic block model [Holland-Laskey-Leinhardt ’83]
Planted partition model [Condon-Karp 01’]
p = 0.8 q = 0.09
Jiaming Xu (Purdue) SDP for community detection 4
![Page 8: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/8.jpg)
Stochastic block model [Holland-Laskey-Leinhardt ’83]
Planted partition model [Condon-Karp 01’]
p = 0.8 q = 0.09
Jiaming Xu (Purdue) SDP for community detection 5
![Page 9: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/9.jpg)
Stochastic block model [Holland-Laskey-Leinhardt ’83]
Planted partition model [Condon-Karp 01’]
p = 0.8 q = 0.09
Jiaming Xu (Purdue) SDP for community detection 5
![Page 10: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/10.jpg)
SBM - adjacency matrix view
p
p
p
q
q
• n: total number of nodes
• k: number of communities
• p: within-community edge prob. q: across-community edge prob.
Jiaming Xu (Purdue) SDP for community detection 6
![Page 11: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/11.jpg)
Exact recovery
C∗ −→ A −→ C
• Goal: exact recovery
PC = C∗ n→∞−−−→ 1
• AlternativesI almost exact recovery:
[Mossel-Neeman-Sly ’14, Abbe-Sandon ’15, Montanari ’15,Zhang-Zhou’15, Yun-Proutiere ’15]...
I correlated recovery:[Decelle-Krzakala-Moore-Zdeborova ’11, Mossel-Neeman-Sly ’12 ’13,Massoulie ’13]...
Jiaming Xu (Purdue) SDP for community detection 7
![Page 12: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/12.jpg)
Objectives of this talk
Exact recovery:
PC = C∗ n→∞−−−→ 1
• Information limit: When is exact recovery possible (impossible)?
• Is the information limit achievable in polynomial time, e.g., viasemidefinite programming?
Jiaming Xu (Purdue) SDP for community detection 8
![Page 13: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/13.jpg)
Remainder of the talk
1 Two equal-sized communities
2 Multiple equal-sized communities
3 Conclusions
Jiaming Xu (Purdue) SDP for community detection 9
![Page 14: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/14.jpg)
Two equal-sized communities: Binary symmetric SBM
Model:
• n nodes partitioned into two communities of size n2 (σ∗i = ±1).
• i ∼ j independently w.p.
p = a logn
n σ∗i = σ∗jq = b logn
n σ∗i 6= σ∗j
Remarks
• a+ b > 2 is the connectivity threshold and necessary for exactrecovery
Jiaming Xu (Purdue) SDP for community detection 10
![Page 15: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/15.jpg)
Two equal-sized communities: Binary symmetric SBM
Model:
• n nodes partitioned into two communities of size n2 (σ∗i = ±1).
• i ∼ j independently w.p.
p = a logn
n σ∗i = σ∗jq = b logn
n σ∗i 6= σ∗jRemarks
• a+ b > 2 is the connectivity threshold and necessary for exactrecovery
Jiaming Xu (Purdue) SDP for community detection 10
![Page 16: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/16.jpg)
Two equal-sized communities: MLE ⇒ SDP relaxation
• Maximum likelihood estimator (MLE): Assume p ≥ q
maxσ〈A, σσ>〉 → # of in-cluster edges
s.t. σi ∈ ±1 i ∈ [n]
σ>1 = 0
lift: Y=σσ>========⇒ max
Y〈A, Y 〉
s.t.
Yii = 1 i ∈ [n]
〈J, Y 〉 = 0
• Goal: P
YSDP =
−1
−11
1
→ 1
Jiaming Xu (Purdue) SDP for community detection 11
![Page 17: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/17.jpg)
Two equal-sized communities: MLE ⇒ SDP relaxation
• Maximum likelihood estimator (MLE): Assume p ≥ q
maxσ〈A, σσ>〉 → # of in-cluster edges
s.t. σi ∈ ±1 i ∈ [n]
σ>1 = 0
lift: Y=σσ>========⇒ max
Y〈A, Y 〉
s.t. rank(Y ) = 1
Yii = 1 i ∈ [n]
〈J, Y 〉 = 0
• Goal: P
YSDP =
−1
−11
1
→ 1
Jiaming Xu (Purdue) SDP for community detection 11
![Page 18: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/18.jpg)
Two equal-sized communities: MLE ⇒ SDP relaxation
• Maximum likelihood estimator (MLE): Assume p ≥ q
maxσ〈A, σσ>〉 → # of in-cluster edges
s.t. σi ∈ ±1 i ∈ [n]
σ>1 = 0
lift: Y=σσ>========⇒ max
Y〈A, Y 〉
s.t. Y 0
Yii = 1 i ∈ [n]
〈J, Y 〉 = 0
• Goal: P
YSDP =
−1
−11
1
→ 1
Jiaming Xu (Purdue) SDP for community detection 11
![Page 19: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/19.jpg)
Two equal-sized communities: MLE ⇒ SDP relaxation
• Maximum likelihood estimator (MLE): Assume p ≥ q
maxσ〈A, σσ>〉 → # of in-cluster edges
s.t. σi ∈ ±1 i ∈ [n]
σ>1 = 0
lift: Y=σσ>========⇒ max
Y〈A, Y 〉
s.t. Y 0
Yii = 1 i ∈ [n]
〈J, Y 〉 = 0
• Goal: P
YSDP =
−1
−11
1
→ 1
Jiaming Xu (Purdue) SDP for community detection 11
![Page 20: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/20.jpg)
Two equal-sized communities: Optimal recovery via SDP
Theorem (Abbe-Bandeira-Hall ’14, Mossel-Neeman-Sly ’14)
For two equal-sized communities with p = a log n/n and q = b log n/n:
• If (√a−√b)2 > 2, recovery is achievable in polynomial-time.
• If (√a−√b)2 < 2, recovery is impossible.
Theorem (Hajek-Wu-X. ’14)
SDP achieves the optimal recovery threshold (√a−√b)2 > 2.
Remarks
• originally conjectured in [Abbe-Bandeira-Hall ’14]
• independently proved by [Bandeira ’15]
• P
YSDP =
−1
−11
1
= 1− n−Ω(1)
Jiaming Xu (Purdue) SDP for community detection 12
![Page 21: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/21.jpg)
Two equal-sized communities: Optimal recovery via SDP
Theorem (Abbe-Bandeira-Hall ’14, Mossel-Neeman-Sly ’14)
For two equal-sized communities with p = a log n/n and q = b log n/n:
• If (√a−√b)2 > 2, recovery is achievable in polynomial-time.
• If (√a−√b)2 < 2, recovery is impossible.
Theorem (Hajek-Wu-X. ’14)
SDP achieves the optimal recovery threshold (√a−√b)2 > 2.
Remarks
• originally conjectured in [Abbe-Bandeira-Hall ’14]
• independently proved by [Bandeira ’15]
• P
YSDP =
−1
−11
1
= 1− n−Ω(1)
Jiaming Xu (Purdue) SDP for community detection 12
![Page 22: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/22.jpg)
Two equal-sized communities: Optimal recovery via SDP
Theorem (Abbe-Bandeira-Hall ’14, Mossel-Neeman-Sly ’14)
For two equal-sized communities with p = a log n/n and q = b log n/n:
• If (√a−√b)2 > 2, recovery is achievable in polynomial-time.
• If (√a−√b)2 < 2, recovery is impossible.
Theorem (Hajek-Wu-X. ’14)
SDP achieves the optimal recovery threshold (√a−√b)2 > 2.
Remarks
• originally conjectured in [Abbe-Bandeira-Hall ’14]
• independently proved by [Bandeira ’15]
• P
YSDP =
−1
−11
1
= 1− n−Ω(1)
Jiaming Xu (Purdue) SDP for community detection 12
![Page 23: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/23.jpg)
Two equal-sized communities: Dual certificate argument
YSDP = arg maxY〈A, Y 〉
dual variables
s.t. Y 0
S 0
Yii = 1
D = diag di
〈J, Y 〉 = 0
λ ∈ R
• di = (# of nbrs in own cluster)− (# of nbrs in other cluster)∼ Binom(n/2− 1, p)− Binom(n/2, q)
• S = D −A+ λJ 0 if λ ≥ (p+ q)/2 and min di ≥ ‖A− E [A] ‖• min di = ΩP (log n) if
√a−√b >√
2
• ‖A− E [A] ‖ = OP (√
log n): 2nd-order stochastic dominance[Tomozei-Massoulie ’14] + result for iid matrix [Seginer ’00]
Jiaming Xu (Purdue) SDP for community detection 13
![Page 24: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/24.jpg)
Two equal-sized communities: Dual certificate argument
YSDP = arg maxY〈A, Y 〉 dual variables
s.t. Y 0 S 0
Yii = 1 D = diag di〈J, Y 〉 = 0 λ ∈ R
• di = (# of nbrs in own cluster)− (# of nbrs in other cluster)∼ Binom(n/2− 1, p)− Binom(n/2, q)
• S = D −A+ λJ 0 if λ ≥ (p+ q)/2 and min di ≥ ‖A− E [A] ‖• min di = ΩP (log n) if
√a−√b >√
2
• ‖A− E [A] ‖ = OP (√
log n): 2nd-order stochastic dominance[Tomozei-Massoulie ’14] + result for iid matrix [Seginer ’00]
Jiaming Xu (Purdue) SDP for community detection 13
![Page 25: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/25.jpg)
Two equal-sized communities: Dual certificate argument
YSDP = arg maxY〈A, Y 〉 dual variables
s.t. Y 0 S 0
Yii = 1 D = diag di〈J, Y 〉 = 0 λ ∈ R
• di = (# of nbrs in own cluster)− (# of nbrs in other cluster)∼ Binom(n/2− 1, p)− Binom(n/2, q)
• S = D −A+ λJ 0 if λ ≥ (p+ q)/2 and min di ≥ ‖A− E [A] ‖• min di = ΩP (log n) if
√a−√b >√
2
• ‖A− E [A] ‖ = OP (√
log n): 2nd-order stochastic dominance[Tomozei-Massoulie ’14] + result for iid matrix [Seginer ’00]
Jiaming Xu (Purdue) SDP for community detection 13
![Page 26: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/26.jpg)
Two equal-sized communities: Dual certificate argument
YSDP = arg maxY〈A, Y 〉 dual variables
s.t. Y 0 S 0
Yii = 1 D = diag di〈J, Y 〉 = 0 λ ∈ R
• di = (# of nbrs in own cluster)− (# of nbrs in other cluster)∼ Binom(n/2− 1, p)− Binom(n/2, q)
• S = D −A+ λJ 0 if λ ≥ (p+ q)/2 and min di ≥ ‖A− E [A] ‖
• min di = ΩP (log n) if√a−√b >√
2
• ‖A− E [A] ‖ = OP (√
log n): 2nd-order stochastic dominance[Tomozei-Massoulie ’14] + result for iid matrix [Seginer ’00]
Jiaming Xu (Purdue) SDP for community detection 13
![Page 27: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/27.jpg)
Two equal-sized communities: Dual certificate argument
YSDP = arg maxY〈A, Y 〉 dual variables
s.t. Y 0 S 0
Yii = 1 D = diag di〈J, Y 〉 = 0 λ ∈ R
• di = (# of nbrs in own cluster)− (# of nbrs in other cluster)∼ Binom(n/2− 1, p)− Binom(n/2, q)
• S = D −A+ λJ 0 if λ ≥ (p+ q)/2 and min di ≥ ‖A− E [A] ‖• min di = ΩP (log n) if
√a−√b >√
2
• ‖A− E [A] ‖ = OP (√
log n): 2nd-order stochastic dominance[Tomozei-Massoulie ’14] + result for iid matrix [Seginer ’00]
Jiaming Xu (Purdue) SDP for community detection 13
![Page 28: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/28.jpg)
Two equal-sized communities: Dual certificate argument
YSDP = arg maxY〈A, Y 〉 dual variables
s.t. Y 0 S 0
Yii = 1 D = diag di〈J, Y 〉 = 0 λ ∈ R
• di = (# of nbrs in own cluster)− (# of nbrs in other cluster)∼ Binom(n/2− 1, p)− Binom(n/2, q)
• S = D −A+ λJ 0 if λ ≥ (p+ q)/2 and min di ≥ ‖A− E [A] ‖• min di = ΩP (log n) if
√a−√b >√
2
• ‖A− E [A] ‖ = OP (√
log n): 2nd-order stochastic dominance[Tomozei-Massoulie ’14] + result for iid matrix [Seginer ’00]
Jiaming Xu (Purdue) SDP for community detection 13
![Page 29: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/29.jpg)
k equal-sized communities: MLE ⇒ SDP relaxation
max
k∑`=1
〈A,θ`θ>` 〉
max 〈A,Z〉
s.t. θ` ∈ 0, 1n
lift: Z=∑k
`=1 θ`θ>`⇐===========⇒ s.t.
〈θ`,1〉 = n/k
Zii = 1 ∀i ∈ [n]
〈θ`,θ`′〉 = 0, ` 6= `′
Zij ≥ 0,∑j
Zij = n/k
Goal: P
ZSDP =
11
11
0
0
→ 1
Jiaming Xu (Purdue) SDP for community detection 14
![Page 30: Semidefinite Programming Relaxations for Recovering …jx77/Jiaming-fudan.pdfTwo equal-sized communities: MLE )SDP relaxation Maximum likelihood estimator (MLE): Assume p q max ...](https://reader035.fdocuments.in/reader035/viewer/2022081621/611d846970a84528c675736f/html5/thumbnails/30.jpg)
k equal-sized communities: MLE ⇒ SDP relaxation
max
k∑`=1
〈A,θ`θ>` 〉 max 〈A,Z〉
s.t. θ` ∈ 0, 1nlift: Z=
∑k`=1 θ`θ
>`⇐===========⇒ s.t. rank(Z) = k
〈θ`,1〉 = n/k Zii = 1 ∀i ∈ [n]
〈θ`,θ`′〉 = 0, ` 6= `′ Zij ≥ 0,∑j
Zij = n/k
Goal: P
ZSDP =
11
11
0
0
→ 1
Jiaming Xu (Purdue) SDP for community detection 14
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k equal-sized communities: MLE ⇒ SDP relaxation
max
k∑`=1
〈A,θ`θ>` 〉 max 〈A,Z〉
s.t. θ` ∈ 0, 1nlift: Z=
∑k`=1 θ`θ
>`⇐===========⇒ s.t. Z 0
〈θ`,1〉 = n/k Zii = 1 ∀i ∈ [n]
〈θ`,θ`′〉 = 0, ` 6= `′ Zij ≥ 0,∑j
Zij = n/k
Goal: P
ZSDP =
11
11
0
0
→ 1
Jiaming Xu (Purdue) SDP for community detection 14
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k equal-sized communities: MLE ⇒ SDP relaxation
max
k∑`=1
〈A,θ`θ>` 〉 max 〈A,Z〉
s.t. θ` ∈ 0, 1nlift: Z=
∑k`=1 θ`θ
>`⇐===========⇒ s.t. Z 0
〈θ`,1〉 = n/k Zii = 1 ∀i ∈ [n]
〈θ`,θ`′〉 = 0, ` 6= `′ Zij ≥ 0,∑j
Zij = n/k
Goal: P
ZSDP =
11
11
0
0
→ 1
Jiaming Xu (Purdue) SDP for community detection 14
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k equal-sized communities: optimal recovery via SDP
Theorem (Hajek-Wu-X. ’15)
For a fixed k communities with p = a log n/n and q = b log n/n.
• If√a−√b >√k, exact recovery is attained via SDP in poly-time.
• If√a−√b <√k, exact recovery is impossible.
Remarks
• Extended to k = o(log n) in [Agarwal-Bandeira-Koiliaris-Kolla ’15]
• Extended to the case with multiple unequal-sized clusters[Perry-Wein ’15]
• Heterogeneous setting: [Yun-Proutiere ’14] and [Abbe-Sandon ’15]
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k equal-sized communities: optimal recovery via SDP
Theorem (Hajek-Wu-X. ’15)
For a fixed k communities with p = a log n/n and q = b log n/n.
• If√a−√b >√k, exact recovery is attained via SDP in poly-time.
• If√a−√b <√k, exact recovery is impossible.
Remarks
• Extended to k = o(log n) in [Agarwal-Bandeira-Koiliaris-Kolla ’15]
• Extended to the case with multiple unequal-sized clusters[Perry-Wein ’15]
• Heterogeneous setting: [Yun-Proutiere ’14] and [Abbe-Sandon ’15]
Jiaming Xu (Purdue) SDP for community detection 15
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When does SDP cease to be optimal?
Theorem (Hajek-Wu-X. ’COLT16)
• If k log n, SDP achieves the optimal exact recovery threshold.
• If k ≥ c log n, SDP is suboptimal by a constant factor.
• If k log n, SDP is order-suboptimal.
Remarks
• A “hard but informationally possible“ regime is conjectured to existfor exact recovery when k log n [Chen-X. ’14]
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Concluding remarks
1
12/3
p = cq = Θ(n−α)
s = Θ(nβ)
1/2
impossible
easy
1/2hard
spectral condition
O α
β
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References
• B. Hajek, Y. Wu & J. X. Achieving exact cluster recovery threshold viasemidefinite programming. (Transactions on IT ’16)
• B. Hajek, Y. Wu & J. X. Achieving exact cluster recovery threshold viasemidefinite programming: Extensions. (Transactions on IT ’16)
• B. Hajek, Y. Wu & J. X. Semidefinite programs for exact recovery of a
hidden community. (COLT’16)
SDP in real networks
• Y. Chen, X. Li, and J. X. (2015), Convexified modularity maximization fordegree-corrected stochastic block models. arXiv:1512.08425.
• Code available at http://people.orie.cornell.edu/yudong.chen/cmm
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