### Abstract

In this paper we establish links between, and new results for, three problems that are not usually considered together. The first is a matrix decomposition problem that arises in areas such as statistical modeling and signal processing: given a matrix X formed as the sum of an unknown diagonal matrix and an unknown low-rank positive semidefinite matrix, decompose X into these constituents. The second problem we consider is to determine the facial structure of the set of correlation matrices, a convex set also known as the elliptope. This convex body, and particularly its facial structure, plays a role in applications from combinatorial optimization to mathematical finance. The third problem is a basic geometric question: given points *v*_{1}, *v*_{2}, . . . , *v*_{n} ∈ **R**^{k} (where *n* > *k*) determine whether there is a centered ellipsoid passing *exactly* through all the points. We show that in a precise sense these three problems are equivalent. Furthermore we establish a simple sufficient condition on a subspace *U* that ensures any positive semidefinite matrix *L* with column space *U* can be recovered from *D* + *L* for any diagonal matrix *D* using a convex optimization-based heuristic known as minimum trace factor analysis. This result leads to a new understanding of the structure of rank-deficient correlation matrices and a simple condition on a set of points that ensures there is a centered ellipsoid passing through them.

Original language | English |
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Pages (from-to) | 1395-1416 |

Number of pages | 22 |

Journal | SIAM Journal on Matrix Analysis and Applications |

Volume | 33 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2012 |

Externally published | Yes |

### Keywords

- Elliptope
- Frisch scheme
- Minimum trace factor analysis
- Semidefinite programming
- Subspace coherence

## Cite this

*SIAM Journal on Matrix Analysis and Applications*,

*33*(4), 1395-1416. https://doi.org/10.1137/120872516