Diagonal and low-rank decompositions and fitting ellipsoids to random points
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Parrilo_Diagonal and.pdf
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Author(s)
Saunderson, James F.
Parrilo, Pablo A.
Willsky, Alan S.
Date Issued
December 2013
Journal
52nd IEEE Conference on Decision and Control
Publisher
Institute of Electrical and Electronics Engineers (IEEE)
Citation
Saunderson, James, Pablo A. Parrilo, and Alan S. Willsky. “Diagonal and Low-Rank Decompositions and Fitting Ellipsoids to Random Points.” 52nd IEEE Conference on Decision and Control (12-13 December 2013), Firenze, Italy. IEEE. p.6031-6036.
Version
Author's final manuscript
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[superscript β] for β ϵ (5/6, 1) and some constant c.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
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DOI of Published Version
https://doi.org/10.1109/CDC.2013.6760842
