Exact clustering of weighted graphs via semidefinite programming
(2019) In Journal of Machine Learning Research 20. p.1-34- Abstract
As a model problem for clustering, we consider the densest k-disjoint-clique problem of partitioning a weighted complete graph into k disjoint subgraphs such that the sum of the densities of these subgraphs is maximized. We establish that such subgraphs can be recovered from the solution of a particular semidefinite relaxation with high probability if the input graph is sampled from a distribution of clusterable graphs. Specifically, the semidefinite relaxation is exact if the graph consists of k large disjoint subgraphs, corresponding to clusters, with weight concentrated within these subgraphs, plus a moderate number of nodes not belonging to any cluster. Further, we establish that if noise is weakly obscuring these clusters, i.e, the... (More)
As a model problem for clustering, we consider the densest k-disjoint-clique problem of partitioning a weighted complete graph into k disjoint subgraphs such that the sum of the densities of these subgraphs is maximized. We establish that such subgraphs can be recovered from the solution of a particular semidefinite relaxation with high probability if the input graph is sampled from a distribution of clusterable graphs. Specifically, the semidefinite relaxation is exact if the graph consists of k large disjoint subgraphs, corresponding to clusters, with weight concentrated within these subgraphs, plus a moderate number of nodes not belonging to any cluster. Further, we establish that if noise is weakly obscuring these clusters, i.e, the between-cluster edges are assigned very small weights, then we can recover significantly smaller clusters. For example, we show that in approximately sparse graphs, where the between-cluster weights tend to zero as the size n of the graph tends to infinity, we can recover clusters of size polylogarithmic in n under certain conditions on the distribution of edge weights. Empirical evidence from numerical simulations is also provided to support these theoretical phase transitions to perfect recovery of the cluster structure.
(Less)
- author
- Pirinen, Aleksis LU and Ames, Brendan
- organization
- publishing date
- 2019
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Clustering, Densest subgraph, Semidefinite programming, Sparse graphs, Stochastic block models
- in
- Journal of Machine Learning Research
- volume
- 20
- pages
- 1 - 34
- publisher
- Microtome Publishing
- external identifiers
-
- scopus:85072637509
- ISSN
- 1532-4435
- language
- English
- LU publication?
- yes
- id
- 0d20fe28-6914-4ad9-8eba-65a0620181f8
- alternative location
- http://jmlr.org/papers/v20/16-128.html
- date added to LUP
- 2019-10-07 13:51:31
- date last changed
- 2022-04-18 18:24:46
@article{0d20fe28-6914-4ad9-8eba-65a0620181f8,
abstract = {{<p>As a model problem for clustering, we consider the densest k-disjoint-clique problem of partitioning a weighted complete graph into k disjoint subgraphs such that the sum of the densities of these subgraphs is maximized. We establish that such subgraphs can be recovered from the solution of a particular semidefinite relaxation with high probability if the input graph is sampled from a distribution of clusterable graphs. Specifically, the semidefinite relaxation is exact if the graph consists of k large disjoint subgraphs, corresponding to clusters, with weight concentrated within these subgraphs, plus a moderate number of nodes not belonging to any cluster. Further, we establish that if noise is weakly obscuring these clusters, i.e, the between-cluster edges are assigned very small weights, then we can recover significantly smaller clusters. For example, we show that in approximately sparse graphs, where the between-cluster weights tend to zero as the size n of the graph tends to infinity, we can recover clusters of size polylogarithmic in n under certain conditions on the distribution of edge weights. Empirical evidence from numerical simulations is also provided to support these theoretical phase transitions to perfect recovery of the cluster structure.</p>}},
author = {{Pirinen, Aleksis and Ames, Brendan}},
issn = {{1532-4435}},
keywords = {{Clustering; Densest subgraph; Semidefinite programming; Sparse graphs; Stochastic block models}},
language = {{eng}},
pages = {{1--34}},
publisher = {{Microtome Publishing}},
series = {{Journal of Machine Learning Research}},
title = {{Exact clustering of weighted graphs via semidefinite programming}},
url = {{http://jmlr.org/papers/v20/16-128.html}},
volume = {{20}},
year = {{2019}},
}