Abstract
We study a semidefinite programming (SDP) relaxation of the maximum likelihood estimation for exactly recovering a hidden community of cardinality K from an n × n symmetric data matrix A, where for distinct indices i, j, Aij ∼ P if i, j are both in the community and Aij ∼ Q otherwise, for two known probability distributions P and Q. We identify a sufficient condition and a necessary condition for the success of SDP for the general model. For both the Bernoulli case (P = Bern(p) and Q = Bern(q) with p > q) and the Gaussian case (P = N(µ, 1) and Q = N(0, 1) with µ > 0), which correspond to the problem of planted dense subgraph recovery and submatrix localization respectively, the general results lead to the following findings: (1) If K = ω(n/log n), SDP attains the information-theoretic recovery limits with sharp constants; (2) If K = Θ(n/log n), SDP is order-wise optimal, but strictly suboptimal by a constant factor; (3) If K = o(n/log n) and K → ∞, SDP is order-wise suboptimal. The same critical scaling for K is found to hold, up to constant factors, for the performance of SDP on the stochastic block model of n vertices partitioned into multiple communities of equal size K. A key ingredient in the proof of the necessary condition is a construction of a primal feasible solution based on random perturbation of the true cluster matrix.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 1051-1095 |
| Number of pages | 45 |
| Journal | Journal of Machine Learning Research |
| Volume | 49 |
| Issue number | June |
| State | Published - Jun 6 2016 |
| Event | 29th Conference on Learning Theory, COLT 2016 - New York, United States Duration: Jun 23 2016 → Jun 26 2016 |
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Keywords
- Planted dense subgraph recovery
- Semidefinite programming relaxations
- Stochastic block model
- Submatrix localization
ASJC Scopus subject areas
- Software
- Control and Systems Engineering
- Statistics and Probability
- Artificial Intelligence
Cite this
Semidefinite programs for exact recovery of a hidden community. / Hajek, Bruce; Wu, Yihong; Xu, Jiaming.
In: Journal of Machine Learning Research, Vol. 49, No. June, 06.06.2016, p. 1051-1095.Research output: Contribution to journal › Conference article
}
TY - JOUR
T1 - Semidefinite programs for exact recovery of a hidden community
AU - Hajek, Bruce
AU - Wu, Yihong
AU - Xu, Jiaming
PY - 2016/6/6
Y1 - 2016/6/6
N2 - We study a semidefinite programming (SDP) relaxation of the maximum likelihood estimation for exactly recovering a hidden community of cardinality K from an n × n symmetric data matrix A, where for distinct indices i, j, Aij ∼ P if i, j are both in the community and Aij ∼ Q otherwise, for two known probability distributions P and Q. We identify a sufficient condition and a necessary condition for the success of SDP for the general model. For both the Bernoulli case (P = Bern(p) and Q = Bern(q) with p > q) and the Gaussian case (P = N(µ, 1) and Q = N(0, 1) with µ > 0), which correspond to the problem of planted dense subgraph recovery and submatrix localization respectively, the general results lead to the following findings: (1) If K = ω(n/log n), SDP attains the information-theoretic recovery limits with sharp constants; (2) If K = Θ(n/log n), SDP is order-wise optimal, but strictly suboptimal by a constant factor; (3) If K = o(n/log n) and K → ∞, SDP is order-wise suboptimal. The same critical scaling for K is found to hold, up to constant factors, for the performance of SDP on the stochastic block model of n vertices partitioned into multiple communities of equal size K. A key ingredient in the proof of the necessary condition is a construction of a primal feasible solution based on random perturbation of the true cluster matrix.
AB - We study a semidefinite programming (SDP) relaxation of the maximum likelihood estimation for exactly recovering a hidden community of cardinality K from an n × n symmetric data matrix A, where for distinct indices i, j, Aij ∼ P if i, j are both in the community and Aij ∼ Q otherwise, for two known probability distributions P and Q. We identify a sufficient condition and a necessary condition for the success of SDP for the general model. For both the Bernoulli case (P = Bern(p) and Q = Bern(q) with p > q) and the Gaussian case (P = N(µ, 1) and Q = N(0, 1) with µ > 0), which correspond to the problem of planted dense subgraph recovery and submatrix localization respectively, the general results lead to the following findings: (1) If K = ω(n/log n), SDP attains the information-theoretic recovery limits with sharp constants; (2) If K = Θ(n/log n), SDP is order-wise optimal, but strictly suboptimal by a constant factor; (3) If K = o(n/log n) and K → ∞, SDP is order-wise suboptimal. The same critical scaling for K is found to hold, up to constant factors, for the performance of SDP on the stochastic block model of n vertices partitioned into multiple communities of equal size K. A key ingredient in the proof of the necessary condition is a construction of a primal feasible solution based on random perturbation of the true cluster matrix.
KW - Planted dense subgraph recovery
KW - Semidefinite programming relaxations
KW - Stochastic block model
KW - Submatrix localization
UR - http://www.scopus.com/inward/record.url?scp=85072253438&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85072253438&partnerID=8YFLogxK
M3 - Conference article
AN - SCOPUS:85072253438
VL - 49
SP - 1051
EP - 1095
JO - Journal of Machine Learning Research
JF - Journal of Machine Learning Research
SN - 1532-4435
IS - June
ER -