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## The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices (2011)

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7509 |
Matrix Analysis
- Horn, Johnson
- 1985
(Show Context)
Citation Context ... M.I.T. We gratefully acknowledge the Singapore-MIT alliance for funding F.B.G’s stay. 12 FLORENT BENAYCH-GEORGES AND RAJ RAO NADAKUDITI In this scenario, one can use Weyl’s interlacing inequalities =-=[20]-=- to obtain coarse bounds for the eigenvalues of the sum in terms of the eigenvalues of Xn. When the norm of Pn is small relative to the norm of Xn, tools from perturbation theory (see [20, Chapter 6] ... |

579 | The Concentration of Measure Phenomenon - Ledoux - 2001 |

440 | Spectral analysis of large dimensional random matrices - Bai, Silverstein - 2010 |

422 | Free random variables - Voiculescu, Dykema, et al. - 1992 |

340 | Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory - Deift, Kriecherbauer, et al. - 1999 |

308 |
On the distribution of the roots of certain symmetric matrices
- Wigner
- 1958
(Show Context)
Citation Context ...n be any probability measure. Consequently, the aforementioned results in the literature can be rederived rather simply using the formulas in Section 2 by substituting µX with the semi-circle measure =-=[35]-=- (for Gaussian matrices), the Marčenko-Pastur measure [23] (for Wishart matrices) or the free Jacobi measure (for Jacobi matrices [14]). See Section 3 for some concrete computations. The development o... |

287 |
The semicircle law, free random variables and entropy
- Hiai, Petz
- 2000
(Show Context)
Citation Context ...rns out that these G, T and D integral transforms that emerge in the respective additive, multiplicative and rectangular cases are deeply related to very well known objects in free probability theory =-=[34, 19]-=- that linearize (non-commutative) free additive, multiplicative [34] and rectangular [9, 8] convolutions respectively. The emergence of these transforms in the context of the study of the extreme/isol... |

229 | An Introduction to Random Matrices
- Anderson, Guionnet, et al.
- 2009
(Show Context)
Citation Context ...D RAJ RAO NADAKUDITI is not anymore symmetric (or Hermitian), with singular values σ1(Xn) ≥ · · · ≥ σn(Xn). The integer m ≥ n shall also tend to infinity 2 in such a way that n/m tends to a limit c ∈ =-=[0, 1]-=-. Assume that the empirical distribution on the set of its singular values, i.e. n∑ 1 δσj(Xn) n j=1 converges almost surely weakly, as n, m tend to infinity, to a non-random, compactly supported proba... |

181 | Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices
- Baik, Arous, et al.
(Show Context)
Citation Context ...bution of this paper. In doing so, we dramatically extend the results found in the literature for the eigenvalue phase transition in such finite, low rank perturbation models well beyond the Gaussian =-=[3, 4, 28, 21, 17, 13, 6]-=-, Wishart [16, 27, 25] and Jacobi settings [24]. In our situation, the distribution µX in Figure 1 can be any probability measure. Consequently, the aforementioned results in the literature can be red... |

157 | Eigenvalues of large sample covariance matrices of spiked population models
- Baik, Silverstein
(Show Context)
Citation Context ...bution of this paper. In doing so, we dramatically extend the results found in the literature for the eigenvalue phase transition in such finite, low rank perturbation models well beyond the Gaussian =-=[3, 4, 28, 21, 17, 13, 6]-=-, Wishart [16, 27, 25] and Jacobi settings [24]. In our situation, the distribution µX in Figure 1 can be any probability measure. Consequently, the aforementioned results in the literature can be red... |

149 |
A limit theorem for the norm of random matrices
- Geman
- 1980
(Show Context)
Citation Context ...e covariance matrices and the multiplicative case. Let Gn be an n ×m real (or complex) matrix with independent, zero mean, normally distributed entries with variance 1. Let Xn = GnG∗ n/m. It is known =-=[23, 18]-=- that, as n, m −→ ∞ with n/m → c > 0, the spectral measure of Xn converges to the famous Marčenko-Pastur distribution with density dµX(x) := 1 2πcx ( √ (b − x)(x − a) [a,b](x)dx + max 0, 1 − 1 ) δ0 c ... |

119 |
Asymptotics of sample eigenstructure for a large dimensional spiked covariance model
- Paul
- 2007
(Show Context)
Citation Context ...we dramatically extend the results found in the literature for the eigenvalue phase transition in such finite, low rank perturbation models well beyond the Gaussian [3, 4, 28, 21, 17, 13, 6], Wishart =-=[16, 27, 25]-=- and Jacobi settings [24]. In our situation, the distribution µX in Figure 1 can be any probability measure. Consequently, the aforementioned results in the literature can be rederived rather simply u... |

113 | Integration with respect to the Haar measure on unitary, orthogonal and symplectic group
- Collins, Śniady
(Show Context)
Citation Context ... x (n)2 j x (n)2 i x (n)2 j E[u(n)2 i u (n)2 j v (n)2 i v (n)2 j ] j v(n) i v(n) j v(n) k v(n) l ]. v(n) k (E[u (n)8 i ]E[u (n)8 j ]E[v (n)8 i ]E[v (n)8 j ]) 1 4 . } {{ } =E[u (n)8 1 ] (44) Since, by =-=[15]-=-, E[u (n)8 1 ] = O(n −4 ), the conclusion holds. □ 6 Usually, this result is stated for functions which are 1-Lipschitz with respect to the geodesic distance on the sphere d(u, v) = arccos〈u, v〉, but ... |

96 |
Distribution of eigenvalues in certain sets of random matrices
- Marčenko, Pastur
- 1967
(Show Context)
Citation Context ...oned results in the literature can be rederived rather simply using the formulas in Section 2 by substituting µX with the semi-circle measure [35] (for Gaussian matrices), the Marčenko-Pastur measure =-=[23]-=- (for Wishart matrices) or the free Jacobi measure (for Jacobi matrices [14]). See Section 3 for some concrete computations. The development of the eigenvector aspect of the story is another important... |

95 |
Rank-one modification of the symmetric eigenproblem
- Bunch, Nielsen, et al.
- 1978
(Show Context)
Citation Context ...t Xn be an n×n matrix with real eigenvalues λ1(Xn), . . ., λn(Xn) and Pn be an n×n matrix with rank r ≤ n and real eigenvalues θ1, . . .,θr. A fundamental question in matrix analysis is the following =-=[12, 2]-=-: How are the eigenvalues and eigenvectors of Xn + Pn related to the eigenvalues and eigenvectors of Xn and Pn? When Xn and Pn are diagonalized by the same eigenvectors then we have λi(Xn+Pn) = λj(Xn)... |

92 | Analysis of the limiting spectral distribution of large dimensional random matrices - Choi, Silverstein - 1995 |

89 |
Processes with free increments
- Biane
- 1998
(Show Context)
Citation Context ... we would like to highlight. Generally speaking, the eigenvector question has received much less much attention in random matrix theory and in free probability theory despite impressive breakthroughs =-=[11]-=-. A notable exception is the recent body of work onLOW RANK PERTURBATIONS OF LARGE RANDOM MATRICES 3 ũ u (a) Largest eigenvalue ρ > b in blue when θ > θc (b) Associated when θ > θc eigenvector ũ u (c... |

67 | Tracy-Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices. The Annals of Probability, 35(2):663–714
- Karoui
- 2007
(Show Context)
Citation Context ...we dramatically extend the results found in the literature for the eigenvalue phase transition in such finite, low rank perturbation models well beyond the Gaussian [3, 4, 28, 21, 17, 13, 6], Wishart =-=[16, 27, 25]-=- and Jacobi settings [24]. In our situation, the distribution µX in Figure 1 can be any probability measure. Consequently, the aforementioned results in the literature can be rederived rather simply u... |

64 | Finite sample approximation results for principal component analysis : a matrix perturbation approach
- Nadler
(Show Context)
Citation Context ...we dramatically extend the results found in the literature for the eigenvalue phase transition in such finite, low rank perturbation models well beyond the Gaussian [3, 4, 28, 21, 17, 13, 6], Wishart =-=[16, 27, 25]-=- and Jacobi settings [24]. In our situation, the distribution µX in Figure 1 can be any probability measure. Consequently, the aforementioned results in the literature can be rederived rather simply u... |

57 | Product of random projections, Jacobi ensembles and universality problems arising from free probability. Probability theory and related fields
- Collins
- 2005
(Show Context)
Citation Context ...ulas in Section 2 by substituting µX with the semi-circle measure [35] (for Gaussian matrices), the Marčenko-Pastur measure [23] (for Wishart matrices) or the free Jacobi measure (for Jacobi matrices =-=[14]-=-). See Section 3 for some concrete computations. The development of the eigenvector aspect of the story is another important contribution that we would like to highlight. Generally speaking, the eigen... |

53 | The largest eigenvalues of finite rank deformation of large wigner matrices: convergence and nonuniversality of the fluctuations,” The Annals of Probability
- Capitaine, Donati-Martin, et al.
- 2009
(Show Context)
Citation Context ...bution of this paper. In doing so, we dramatically extend the results found in the literature for the eigenvalue phase transition in such finite, low rank perturbation models well beyond the Gaussian =-=[3, 4, 28, 21, 17, 13, 6]-=-, Wishart [16, 27, 25] and Jacobi settings [24]. In our situation, the distribution µX in Figure 1 can be any probability measure. Consequently, the aforementioned results in the literature can be red... |

52 | The largest eigenvalues of small rank perturbations of Hermitian random matrices, Probab. Theory Related Fields 134
- Péché
- 2006
(Show Context)
Citation Context |

50 |
Lectures on the combinatorics of free probability, volume 335
- Nica, Speicher
- 2006
(Show Context)
Citation Context ...or all probability measures µ1, µ2, µ1 ⊞ µ2 is characterized by the fact that Rµ1⊞µ2(z) = Rµ1(z) + Rµ2(z). The coefficients of the series expansion of Rµ(z) are the so called free cumulants of µ (see =-=[26, 1]-=-). 2.4.2. The T-transform and its relation to multiplicative free convolution. Let µ = δ0 be a law with compact support contained in [0, +∞). Let us denote by [a, b] the convex hull of the support of... |

48 | Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields - Kuijlaars, McLaughlin |

44 | The largest eigenvalue of rank one deformation of large wigner matrices
- Féral, Péché
- 2007
(Show Context)
Citation Context |

43 |
Matrix perturbation theory. Computer Science and Scientific Computing
- Stewart, Sun
- 1990
(Show Context)
Citation Context ... obtain coarse bounds for the eigenvalues of the sum in terms of the eigenvalues of Xn. When the norm of Pn is small relative to the norm of Xn, tools from perturbation theory (see [20, Chapter 6] or =-=[33]-=-) can be employed to improve the characterization of the bounded set in which the eigenvalues of the sum must lie. Exploiting any special structure in the matrices allows us to refine these bounds [22... |

42 | Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices, Electron
- Benaych-Georges, Guionnet, et al.
(Show Context)
Citation Context ...tion phenomenon. The underlying methods can and have been adapted to study the extreme singular values and singular vectors of deformations of rectangular random matrices, as well as the fluctuations =-=[10]-=- and the large deviations [11] of our model. The paper is organized as follows. In Section 2, we state the main results and present the integral transforms alluded to above. Section 3 presents some ex... |

41 | Fundamental limit of sample generalized eigenvalue based detection of signals in noise using relatively few signal-bearing and noise-only samples
- Nadakuditi, Silverstein
(Show Context)
Citation Context ...s found in the literature for the eigenvalue phase transition in such finite, low rank perturbation models well beyond the Gaussian [3, 4, 28, 21, 17, 13, 6], Wishart [16, 27, 25] and Jacobi settings =-=[24]-=-. In our situation, the distribution µX in Figure 1 can be any probability measure. Consequently, the aforementioned results in the literature can be rederived rather simply using the formulas in Sect... |

29 | Rectangular random matrices, related convolution. Probab. Theory Related Fields
- Benaych-Georges
- 2009
(Show Context)
Citation Context ...tiplicative and rectangular cases are deeply related to very well known objects in free probability theory [34, 19] that linearize (non-commutative) free additive, multiplicative [34] and rectangular =-=[9, 8]-=- convolutions respectively. The emergence of these transforms in the context of the study of the extreme/isolated eigenvalue behavior should be of independent interest to free probabilists. This justi... |

28 | A Fourier view on the R-transform and related asymptotics of spherical integrals
- Guionnet, Maida
- 2005
(Show Context)
Citation Context ...ent application of concentration inequalities for random vectors uniformly distributed on high dimensional unit spheres4 FLORENT BENAYCH-GEORGES AND RAJ RAO NADAKUDITI (such as the ones appearing in =-=[19, 20]-=-) to these implicit master equation representations. Consequently, our technique is simpler, more general and brings into focus the source of the phase transition phenomenon. The underlying methods ca... |

25 |
Free random variables, volume 1 of CRM Monograph Series
- Voiculescu, Dykema, et al.
- 1992
(Show Context)
Citation Context ...rns out that these G, T and D integral transforms that emerge in the respective additive, multiplicative and rectangular cases are deeply related to very well known objects in free probability theory =-=[34, 19]-=- that linearize (non-commutative) free additive, multiplicative [34] and rectangular [9, 8] convolutions respectively. The emergence of these transforms in the context of the study of the extreme/isol... |

24 | The polynomial method for random matrices
- Rao, Edelman
- 2005
(Show Context)
Citation Context ...inverses can be expressed in closed form. In settings where the transforms are algebraic so that they can be represented as solutions of polynomial equations, the techniques and software developed in =-=[29]-=- can be utilized. In more complicated settings, one will have to resort to numerical techniques. 3.1. Random Gaussian matrices and the square additive case. Let Xn be an n×n symmetric (or Hermitian) m... |

24 | Weak convergence of random functions defined by the eigenvectors of sample covariance matrices - Silverstein - 1990 |

19 | Infinitely divisible distributions for rectangular free convolution: classification and matricial interpretation - Benaych-Georges |

18 |
An Introduction to Random Matrices. Cambridge studies in advanced mathematics
- Anderson, Guionnet, et al.
- 2009
(Show Context)
Citation Context ...stribution µX of Xn. It turns out that the integral transforms that emerge in the respective additive and multiplicative cases are deeply related to very well known objects in free probability theory =-=[37, 21, 1]-=- that linearize free additive and multiplicative convolutions respectively. In a forthcoming paper [12], we consider the analogue of the problem for the extreme singular values of finite rank deformat... |

13 |
Restricted rank modification of the symmetric eigenvalue problem: theoretical considerations
- Arbenz, Gander, et al.
- 1988
(Show Context)
Citation Context ...)matrixwitheigenvaluesλ1(Xn),...,λn(Xn) and Pn be an n×n symmetric (or Hermitian) matrix with rank r ≤ n and non-zero eigenvalues θ1,...,θr. A fundamental question in matrix analysis is the following =-=[13, 2]-=-: How are the eigenvalues and eigenvectors of Xn +Pn related to the eigenvalues and eigenvectors of Xn and Pn? Date: December 27, 2010. 2000 Mathematics Subject Classification. 15A52, 46L54, 60F99. Ke... |

12 | Refined perturbation bounds for eigenvalues of Hermitian and non-Hermitian matrices
- Ipsen, Nadler
- 2009
(Show Context)
Citation Context ...33]) can be employed to improve the characterization of the bounded set in which the eigenvalues of the sum must lie. Exploiting any special structure in the matrices allows us to refine these bounds =-=[22]-=- but this is pretty much as far as the theory goes. Instead of exact answers we have a system of messy, coupled bounds. The eigenvector story is even more convoluted. Surprisingly, adding some randomn... |

10 | Some limit theorems on the eigenvectors of large-dimensional sample covariance matrices
- Silverstein
- 1984
(Show Context)
Citation Context ...ith mean zero and variance one, then upon placing adequate restrictions on the higher order moments, we label the eigenvectors of X as being Haar-like. Formally speaking, following the development in =-=[30, 31, 32]-=-, when U is Haar-like, then for non-random unit norm vector xn, the vector U ∗ xn will be close to uniformly distributed on the unit real (or complex) sphere. Since our proofs rely heavily on the prop... |

8 | On a surprising relation between the Marchenko-Pastur law, rectangular and square free convolutions
- Benaych-Georges
- 2010
(Show Context)
Citation Context ...tiplicative and rectangular cases are deeply related to very well known objects in free probability theory [34, 19] that linearize (non-commutative) free additive, multiplicative [34] and rectangular =-=[9, 8]-=- convolutions respectively. The emergence of these transforms in the context of the study of the extreme/isolated eigenvalue behavior should be of independent interest to free probabilists. This justi... |

8 | Character expansion method for the first order asymptotics of a matrix integral
- Guionnet, Mäıda
(Show Context)
Citation Context ...ent application of concentration inequalities for random vectors uniformly distributed on high dimensional unit spheres4 FLORENT BENAYCH-GEORGES AND RAJ RAO NADAKUDITI (such as the ones appearing in =-=[19, 20]-=-) to these implicit master equation representations. Consequently, our technique is simpler, more general and brings into focus the source of the phase transition phenomenon. The underlying methods ca... |

8 | On the eigenvectors of large-dimensional sample covariance matrices - Silverstein - 1989 |

5 | Eigenvalue separation in some random matrix models
- Bassler, Forrester, et al.
(Show Context)
Citation Context |

4 |
Statistical mechanics of learning multiple orthogonal signals: asymptotic theory and fluctuation effects
- Hoyle, Rattray
(Show Context)
Citation Context |

4 |
Silverstein Weak convergence of random functions defined by the eigenvectors of sample covariance matrices
- W
- 1990
(Show Context)
Citation Context ...ith mean zero and variance one, then upon placing adequate restrictions on the higher order moments, we label the eigenvectors of X as being Haar-like. Formally speaking, following the development in =-=[30, 31, 32]-=-, when U is Haar-like, then for non-random unit norm vector xn, the vector U ∗ xn will be close to uniformly distributed on the unit real (or complex) sphere. Since our proofs rely heavily on the prop... |

4 |
Spectral Analysis
- Bai, Silverstein
- 2009
(Show Context)
Citation Context ...n converges almost surely to the famous Marčenko-Pastur distribution with density dµX(x) := 1 ( √ (b−x)(x−a) [a,b](x)dx+max 0,1− 2πcx 1 ) δ0, c where a = (1 − √ c) 2 and b = (1 + √ c) 2 . It is known =-=[3]-=- that the extreme eigenvalues converge almost surely to the endpoints of this support. Associated with this spectral measure, we have TµX (z) = z −c−1−sgn(z −a)√ (z −a)(z −b) 2c TµX (b+ ) = 1/ √ c, Tµ... |

2 | Rectangular R-transform at the limit of rectangular spherical integrals
- Benaych-Georges
- 2011
(Show Context)
Citation Context ... random matrices. There too, a phase transition occurs at a threshold determined by an integral transform which plays an analogous role in the computation of the rectangular additive free convolution =-=[7, 8, 9]-=-. The emergence of these transforms in the context of the study of the extreme or isolated eigenvalue behavior should be of independent interest to free probabilists. In doing so, we extend the result... |

2 |
Large deviations of extreme eigenvalues of finite rank deformations of deterministic matrices
- Benaych-Georges, Guionnet, et al.
- 2011
(Show Context)
Citation Context ...g methods can and have been adapted to study the extreme singular values and singular vectors of deformations of rectangular random matrices, as well as the fluctuations [10] and the large deviations =-=[11]-=- of our model. The paper is organized as follows. In Section 2, we state the main results and present the integral transforms alluded to above. Section 3 presents some examples. An outline of the proo... |

1 | Measure concentration lecture notes. Available online at http://www.math.lsa.umich.edu/ barvinok/total710.pdf - Barvinok |

1 |
Silverstein On the eigenvectors of large-dimensional sample covariance matrices
- W
- 1989
(Show Context)
Citation Context ...ith mean zero and variance one, then upon placing adequate restrictions on the higher order moments, we label the eigenvectors of X as being Haar-like. Formally speaking, following the development in =-=[30, 31, 32]-=-, when U is Haar-like, then for non-random unit norm vector xn, the vector U ∗ xn will be close to uniformly distributed on the unit real (or complex) sphere. Since our proofs rely heavily on the prop... |

1 |
andO.Zeitouni.An Introduction to Random Matrices.Cambridgestudies in advanced mathematics
- Anderson
- 2009
(Show Context)
Citation Context ...stribution µX of Xn. It turns out that the integral transforms that emerge in the respective additive and multiplicative cases are deeply related to very well known objects in free probability theory =-=[37, 21, 1]-=- that linearize free additive and multiplicative convolutions respectively. In a forthcoming paper [12], we consider the analogue of the problem for the extreme singular values of finite rank deformat... |

1 |
The extremesingularvalues andsingularvectorsoffinite, low rank perturbations of large random rectangular matrices
- Nadakuditi
(Show Context)
Citation Context ...iplicative cases are deeply related to very well known objects in free probability theory [37, 21, 1] that linearize free additive and multiplicative convolutions respectively. In a forthcoming paper =-=[12]-=-, we consider the analogue of the problem for the extreme singular values of finite rank deformations of rectangular random matrices. There too, a phase transition occurs at a threshold determined by ... |

1 |
Silverstein Some limit theorems on the eigenvectors of large-dimensional sample covariance matrices
- W
- 1984
(Show Context)
Citation Context ... then upon placing adequate restrictions on the higher order moments, for non-random unit norm vector xn, the vector U ∗ xn will be close to uniformly distributed on the unit real (or complex) sphere =-=[33, 34, 35]-=-. Since our proofs rely heavily on the properties of unit norm vectors uniformly distributed on the n-sphere, they could possibly be adapted to the setting where the unit norm vectors are close to uni... |

1 |
Silverstein On the eigenvectorsoflarge-dimensionalsamplecovariancematricesJ
- W
- 1989
(Show Context)
Citation Context ... then upon placing adequate restrictions on the higher order moments, for non-random unit norm vector xn, the vector U ∗ xn will be close to uniformly distributed on the unit real (or complex) sphere =-=[33, 34, 35]-=-. Since our proofs rely heavily on the properties of unit norm vectors uniformly distributed on the n-sphere, they could possibly be adapted to the setting where the unit norm vectors are close to uni... |

1 |
The extreme singular values and singular vectors of finite, low rank perturbations of large random rectangular matrices
- Benaych-Georges, Nadakuditi
(Show Context)
Citation Context ...iplicative cases are deeply related to very well known objects in free probability theory [37, 21, 1] that linearize free additive and multiplicative convolutions respectively. In a forthcoming paper =-=[12]-=-, we consider the analogue of the problem for the extreme singular values of finite rank deformations of rectangular random matrices. There too, a phase transition occurs at a threshold determined by ... |

1 | Matrix Perturbation Theory, Comput - Stewart, Sun - 1990 |