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Mathematics > Numerical Analysis

arXiv:1607.00514 (math)
[Submitted on 2 Jul 2016]

Title:Approximate Joint Matrix Triangularization

Authors:Nicolo Colombo, Nikos Vlassis
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Abstract:We consider the problem of approximate joint triangularization of a set of noisy jointly diagonalizable real matrices. Approximate joint triangularizers are commonly used in the estimation of the joint eigenstructure of a set of matrices, with applications in signal processing, linear algebra, and tensor decomposition. By assuming the input matrices to be perturbations of noise-free, simultaneously diagonalizable ground-truth matrices, the approximate joint triangularizers are expected to be perturbations of the exact joint triangularizers of the ground-truth matrices. We provide a priori and a posteriori perturbation bounds on the `distance' between an approximate joint triangularizer and its exact counterpart. The a priori bounds are theoretical inequalities that involve functions of the ground-truth matrices and noise matrices, whereas the a posteriori bounds are given in terms of observable quantities that can be computed from the input matrices. From a practical perspective, the problem of finding the best approximate joint triangularizer of a set of noisy matrices amounts to solving a nonconvex optimization problem. We show that, under a condition on the noise level of the input matrices, it is possible to find a good initial triangularizer such that the solution obtained by any local descent-type algorithm has certain global guarantees. Finally, we discuss the application of approximate joint matrix triangularization to canonical tensor decomposition and we derive novel estimation error bounds.
Comments: 19 pages
Subjects: Numerical Analysis (math.NA); Machine Learning (cs.LG); Machine Learning (stat.ML)
MSC classes: 15A23, 15A42, 15A45, 15B10
Cite as: arXiv:1607.00514 [math.NA]
  (or arXiv:1607.00514v1 [math.NA] for this version)
  https://doi.org/10.48550/arXiv.1607.00514

Submission history

From: Nicolo Colombo [view email]
[v1] Sat, 2 Jul 2016 14:25:58 UTC (20 KB)
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