Estimation of (near) low-rank matrices with noise and high-dimensional scaling

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Estimation of (near) low-rank matrices with noise and high-dimensional scaling

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by Sahand Negahban , Martin J. Wainwright

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95 - 14 self

BibTeX

@MISC{Negahban_estimationof,
    author = {Sahand Negahban and Martin J. Wainwright},
    title = {Estimation of (near) low-rank matrices with noise and high-dimensional scaling},
    year = {}
}

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Abstract

We study an instance of high-dimensional statistical inference in which the goal is to use N noisy observations to estimate a matrix Θ ∗ ∈ R k×p that is assumed to be either exactly low rank, or “near ” low-rank, meaning that it can be well-approximated by a matrix with low rank. We consider an M-estimator based on regularization by the traceornuclearnormovermatrices, andanalyze its performance under high-dimensional scaling. We provide non-asymptotic bounds on the Frobenius norm error that hold for a generalclassofnoisyobservationmodels,and apply to both exactly low-rank and approximately low-rank matrices. We then illustrate their consequences for a number of specific learning models, including low-rank multivariate or multi-task regression, system identification in vector autoregressive processes, and recovery of low-rank matrices from random projections. Simulations show excellent agreement with the high-dimensional scaling of the error predicted by our theory. 1.

Keyphrases

high-dimensional scaling low-rank matrix low rank non-asymptotic bound random projection multi-task regression frobenius norm error system identification excellent agreement high-dimensional statistical inference vector autoregressive process low-rank multivariate noisy observation

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