## Abstract

Identifying a subspace containing signals of interest in additive noise is a basic system identification problem. Under natural assumptions, this problem is known as the Frisch scheme and can be cast as decomposing an *n* × *n* positive definite matrix as the sum of an unknown diagonal matrix (the noise covariance) and an unknown low-rank matrix (the signal covariance). Our focus in this paper is a natural class of random instances, where the low-rank matrix has a uniformly distributed random column space. In this setting we analyze the behavior of a well-known convex optimization-based heuristic for diagonal and low-rank decomposition called minimum trace factor analysis (MTFA). Conditions for the success of MTFA have an appealing geometric reformulation as finding a (convex) ellipsoid that exactly interpolates a given set of n points. Under the random model, the points are chosen according to a Gaussian distribution. Numerical experiments suggest a remarkable threshold phenomenon: if the (random) column space of the *n* × *n* lowrank matrix has codimension as small as 2 √ *n* then with high probability MTFA successfully performs the decomposition task, otherwise it fails with high probability. In this work we provide numerical evidence and prove partial results in this direction, showing that with high probability MTFA recovers such random low-rank matrices of corank at least *cn*^{β} for *β* ∈ (5/6, 1) and some constant *c*.

Original language | English |
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Title of host publication | Proceedings of the 52nd IEEE Conference on Decision and Control, CDC 2013 |

Subtitle of host publication | December 10-13, 2013, Firenze, Italy |

Publisher | IEEE, Institute of Electrical and Electronics Engineers |

Pages | 6031-6036 |

Number of pages | 6 |

ISBN (Electronic) | 9781467357173 |

ISBN (Print) | 9781467357142 |

DOIs | |

Publication status | Published - 2013 |

Externally published | Yes |

Event | IEEE Conference on Decision and Control 2013 - Florence, Italy Duration: 10 Dec 2013 → 13 Dec 2013 Conference number: 52nd https://ieeexplore.ieee.org/xpl/conhome/6749719/proceeding (Proceedings) |

### Conference

Conference | IEEE Conference on Decision and Control 2013 |
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Abbreviated title | CDC 2013 |

Country/Territory | Italy |

City | Florence |

Period | 10/12/13 → 13/12/13 |

Internet address |