Identifying clusters of similar objects in data plays a significant role in a wide range of applications. As a model problem for clustering, we consider the densest k-disjoint-clique problem, whose goal is to identify the collection of k disjoint cliques of a given weighted complete graph maximizing the sum of the densities of the complete subgraphs induced by these cliques. In this paper, we establish conditions ensuring exact recovery of the densest k cliques of a given graph from the optimal solution of a particular semidefinite program. In particular, the semidefinite relaxation is exact for input graphs corresponding to data consisting of k large, distinct clusters and a smaller number of outliers. This approach also yields a semidefinite relaxation for the biclustering problem with similar recovery guarantees. Given a set of objects and a set of features exhibited by these objects, biclustering seeks to simultaneously group the objects and features according to their expression levels. This problem may be posed as partitioning the nodes of a weighted bipartite complete graph such that the sum of the densities of the resulting bipartite complete subgraphs is maximized. As in our analysis of the densest k-disjoint-clique problem, we show that the correct partition of the objects and features can be recovered from the optimal solution of a semidefinite program in the case that the given data consists of several disjoint sets of objects exhibiting similar features. Empirical evidence from numerical experiments supporting these theoretical guarantees is also provided.
Measurement incompatibility and quantum steering via linear programming
Sorbonne Université, CNRS, LIP6（索邦大学）；Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas (Unicamp)（坎皮纳斯州立大学格列布·瓦塔金物理研究所）；Zuse Institute Berlin（柏林齐塞研究所）；Inria, ENS de Lyon, UCBL, LIP（法国国家信息与自动化研究所，里昂高等师范学校，里昂第一大学）；Berlin Institute of Technology（柏林工业大学）
