Rotational Invariant Estimator for General Noisy Matrices

Bun, Joël; Allez, Romain; Bouchaud, Jean‐Philippe; Potters, Marc

doi:10.1109/tit.2016.2616132

Cited by 79 publications

(118 citation statements)

References 49 publications

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“…This result was first obtained in [16,19], and is the counterpart of the Ledoit and Péché result [20] in the context of multiplicative models (see [14,21] for more details). Note that the square overlap is of order N −1 as soon as σ > 0.…”

Section: A Dyson Brownian Motion For the Resolvantsupporting

confidence: 55%

Two short pieces around the Wigner problem

Bouchaud¹,

Potters²

2018

J. Phys. A: Math. Theor.

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We revisit the classic Wigner semi-circle from two different angles. One consists in studying the Stieltjes transform directly on the real axis, which does not converge to a fixed value but follows a Cauchy distribution that depends on the local eigenvalue density. This result was recently proven by Aizenman & Warzel for a wide class of eigenvalue distributions. We shed new light onto their result using a Coulomb gas method. The second angle is to derive a Langevin equation for the full (matrix) resolvent, extending Dyson's Brownian motion framework. The full matrix structure of this equation allows one to recover known results on the overlaps between the eigenvectors of a fixed matrix and its noisy counterpart.

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Section: A Dyson Brownian Motion For the Resolvantsupporting

confidence: 55%

Two short pieces around the Wigner problem

Bouchaud¹,

Potters²

2018

J. Phys. A: Math. Theor.

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“…In regime (2), we are interested in looking at what is the best we can do in this case. A natural (and probably necessary) assumption is rotation invariance [5], as the only information we know about the singular vectors is orthonormality. It is notable that, in this case, our result coincides with the results proposed by Gavish and Donoho [14], where they consider the estimator from another perspective and restrict the estimator to be conservative (see Definition 3 in [14]).…”

Section: Introductionmentioning

confidence: 99%

High dimensional deformed rectangular matrices with applications in matrix denoising

Ding¹

2020

Bernoulli

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We consider the recovery of a low rank M × N matrix S from its noisy observationS in the high dimensional framework when M is comparable to N . We propose two efficient estimators for S under two different regimes. Our analysis relies on the local asymptotics of the eigenstructure of large dimensional rectangular matrices with finite rank perturbation. We derive the convergent limits and rates for the singular values and vectors for such matrices.

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“…To solve this optimization problem, similar to the family of rotational invariant methods [27], we assume that Γ(E) and E share the same eigenvectors. Here we have: (20) In other words, the eigenvectors are fixed while the eigenvalues are free variables in this optimization.…”

Section: Rmt Based Error Cleaningmentioning

confidence: 99%

Improving Power System State Estimation Based on Matrix-Level Cleaning

Yang

Qiu

Chu

et al. 2020

IEEE Trans. Power Syst.

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Power system state estimation is heavily subjected to measurement error, which comes from the noise of measuring instruments, communication noise, and some unclear randomness. Traditional weighted least square (WLS), as the most universal state estimation method, attempts to minimize the residual between measurements and the estimation of measured variables, but it is unable to handle the measurement error. To solve this problem, based on random matrix theory, this paper proposes a data-driven approach to clean measurement error in matrix-level. Our method significantly reduces the negative effect of measurement error, and conducts a two-stage state estimation scheme combined with WLS. In this method, a Hermitian matrix is constructed to establish an invertible relationship between the eigenvalues of measurements and their covariance matrix. Random matrix tools, combined with an optimization scheme, are used to clean measurement error by shrinking the eigenvalues of the covariance matrix. With great robustness and generality, our approach is particularly suitable for large interconnected power grids. Our method has been numerically evaluated using different testing systems, multiple models of measured noise and matrix size ratios.

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Rotational Invariant Estimator for General Noisy Matrices

Cited by 79 publications

References 49 publications

Two short pieces around the Wigner problem

Two short pieces around the Wigner problem

High dimensional deformed rectangular matrices with applications in matrix denoising

Improving Power System State Estimation Based on Matrix-Level Cleaning

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