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

## TL;DR

This paper introduces a novel chordal decomposition approach for rank-constrained SDPs, enabling efficient solutions for problems like subspace clustering by decomposing large rank constraints into smaller ones.

## Contribution

It develops a method to decompose rank constraints in SDPs using chordal sparsity and a re-weighted heuristic, improving computational efficiency for large-scale problems.

## Key findings

- Significant speed-up in solving rank-minimized SDPs.
- Effective application to subspace clustering tasks.
- Preservation of sparsity pattern across iterations.

## Abstract

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.