Chordal Decomposition in Rank Minimized Semidefinite Programs with Applications to Subspace Clustering

Jared Miller, Yang Zheng, Biel Roig-Solvas, Mario Sznaier, Antonis Papachristodoulou
2019 2019 IEEE 58th Conference on Decision and Control (CDC)  
Semidefinite programs (SDPs) often arise in relaxations of some NP-hard problems, and if the solution of the SDP obeys certain rank constraints, the relaxation will be tight. Decomposition methods based on chordal sparsity have already been applied to speed up the solution of sparse SDPs, but methods for dealing with rank constraints are underdeveloped. This paper leverages a minimum rank completion result to decompose the rank constraint on a single large matrix into multiple rank constraints
more » ... e rank constraints on a set of smaller matrices. The re-weighted heuristic is used as a proxy for rank, and the specific form of the heuristic preserves the sparsity pattern between iterations. Implementations of rank-minimized SDPs through interior-point and first-order algorithms are discussed. The problem of subspace clustering is used to demonstrate the computational improvement of the proposed method. 1 J. Miller, B. Roig-Solvas and M. Sznaier are with the ECE
doi:10.1109/cdc40024.2019.9029620 dblp:conf/cdc/Miller0RSP19 fatcat:zbfqbwkxgvh4pgcsyvls4xj2nu