Chordal Decomposition in Rank Minimized Semidefinite Programs with Applications to Subspace Clustering
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 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.
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Publication Date:
NSF-PAR ID:
10176075
Journal Name:
2019 IEEE 58th Conference on Decision and Control (CDC)
Page Range or eLocation-ID:
4916 to 4921
We study the sparsity of the solutions to systems of linear Diophantine equations with and without non-negativity constraints. The sparsity of a solution vector is the number of its nonzero entries, which is referred to as the$$\ell _0$$${\ell }_{0}$-norm of the vector. Our main results are new improved bounds on the minimal$$\ell _0$$${\ell }_{0}$-norm of solutions to systems$$A\varvec{x}=\varvec{b}$$$Ax=b$, where$$A\in \mathbb {Z}^{m\times n}$$$A\in {Z}^{m×n}$,$${\varvec{b}}\in \mathbb {Z}^m$$$b\in {Z}^{m}$and$$\varvec{x}$$$x$is either a general integer vector (lattice case) or a non-negative integer vector (semigroup case). In certain cases, we give polynomial time algorithms for computing solutions with$$\ell _0$$${\ell }_{0}$-norm satisfying the obtained bounds. We show that our bounds are tight. Our bounds can be seen as functions naturally generalizing the rank of a matrix over$$\mathbb {R}$$$R$, to other subdomains such as$$\mathbb {Z}$$$Z$. We show that these new rank-like functions are all NP-hard to compute in general, but polynomial-time computable for fixed number of variables.