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48
Nonasymptotic theory of random matrices: extreme singular values
 PROCEEDINGS OF THE INTERNATIONAL CONGRESS OF MATHEMATICIANS
, 2010
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Sparse principal component analysis and iterative thresholding, The Annals of Statistics 41
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Random matrices: The distribution of the smallest singular values
, 2009
"... Let ξ be a realvalued random variable of mean zero and variance 1. Let Mn(ξ) denote the n × n random matrix whose entries are iid copies of ξ and σn(Mn(ξ)) denote the least singular value of Mn(ξ). The quantity σn(Mn(ξ)) 2 is thus the least eigenvalue of the Wishart matrix MnM ∗ n. We show that ( ..."
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Cited by 47 (8 self)
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Let ξ be a realvalued random variable of mean zero and variance 1. Let Mn(ξ) denote the n × n random matrix whose entries are iid copies of ξ and σn(Mn(ξ)) denote the least singular value of Mn(ξ). The quantity σn(Mn(ξ)) 2 is thus the least eigenvalue of the Wishart matrix MnM ∗ n. We show that (under a finite moment assumption) the probability distribution nσn(Mn(ξ)) 2 is universal in the sense that it does not depend on the distribution of ξ. In particular, it converges to the same limiting distribution as in the special case when ξ is real gaussian. (The limiting distribution was computed explicitly in this case by Edelman.) We also proved a similar result for complexvalued random variables of mean zero, with real and imaginary parts having variance 1/2 and covariance zero. Similar results are also obtained for the joint distribution of the bottom k singular values of Mn(ξ) for any fixed k (or even for k growing as a small power of n) and for rectangular matrices. Our approach is motivated by the general idea of “property testing ” from combinatorics and theoretical computer science. This seems to be a new approach in the study of spectra of random matrices and combines tools from various areas of mathematics
Random covariance matrices: Universality of local statistics of eigenvalues
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Cooperative Spectrum Sensing based on the Limiting Eigenvalue Ratio Distribution in Wishart Matrices
, 902
"... Abstract—Recent advances in random matrix theory have spurred the adoption of eigenvaluebased detection techniques for cooperative spectrum sensing in cognitive radio. Most of such techniques use the ratio between the largest and the smallest eigenvalues of the received signal covariance matrix to ..."
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Cited by 28 (3 self)
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Abstract—Recent advances in random matrix theory have spurred the adoption of eigenvaluebased detection techniques for cooperative spectrum sensing in cognitive radio. Most of such techniques use the ratio between the largest and the smallest eigenvalues of the received signal covariance matrix to infer the presence or absence of the primary signal. The results derived so far in this field are based on asymptotical assumptions, due to the difficulties in characterizing the exact distribution of the eigenvalues ratio. By exploiting a recent result on the limiting distribution of the smallest eigenvalue in complex Wishart matrices, in this paper we derive an expression for the limiting eigenvalue ratio distribution, which turns out to be much more accurate than the previous approximations also in the nonasymptotical region. This result is then straightforwardly applied to calculate the decision threshold as a function of a target probability of false alarm. Numerical simulations show that the proposed detection rule provides a substantial performance improvement compared to the other eigenvaluebased algorithms. I.
Delocalization and diffusion profile for random band matrices
, 2014
"... We consider Hermitian and symmetric random band matrices H = (hxy) in d> 1 dimensions. The matrix entries hxy, indexed by x, y ∈ (Z/LZ)d, are independent, centred random variables with variances sxy = Ehxy2. We assume that sxy is negligible if x − y  exceeds the band width W. In one dimension ..."
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Cited by 21 (4 self)
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We consider Hermitian and symmetric random band matrices H = (hxy) in d> 1 dimensions. The matrix entries hxy, indexed by x, y ∈ (Z/LZ)d, are independent, centred random variables with variances sxy = Ehxy2. We assume that sxy is negligible if x − y  exceeds the band width W. In one dimension we prove that the eigenvectors of H are delocalized if W L4/5. We also show that the magnitude of the matrix entries Gxy2 of the resolvent G = G(z) = (H − z)−1 is selfaveraging and we compute EGxy2. We show that, as L → ∞ and W L4/5, the behaviour of EGxy2 is governed by a diffusion operator whose diffusion constant we compute. Similar results are obtained in higher dimensions.
Spectral norm of products of random and deterministic matrices
"... Abstract. We study the spectral norm of matrices M that can be factored as M = BA, where A is a random matrix with independent mean zero entries and B is a fixed matrix. Under the (4 + ε)th moment assumption on the entries of A, we show that the spectral norm of such an m×n matrix M is bounded by √ ..."
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Abstract. We study the spectral norm of matrices M that can be factored as M = BA, where A is a random matrix with independent mean zero entries and B is a fixed matrix. Under the (4 + ε)th moment assumption on the entries of A, we show that the spectral norm of such an m×n matrix M is bounded by √ m + √ n, which is sharp. In other words, in regard to the spectral norm, products of random and deterministic matrices behave similarly to random matrices with independent entries. This result along with the previous work of M. Rudelson and the author implies that the smallest singular value of a random m × n matrix with i.i.d. mean zero entries and bounded (4 + ε)th moment is bounded below by √ m − √ n − 1 with high probability. 1.
Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model
, 2010
"... We consider Hermitian and symmetric random band matrices H in d � 1 dimensions. The matrix elements Hxy, indexed by x, y ∈ Λ ⊂ Z d, are independent, uniformly distributed random variables if x − y  is less than the band width W, and zero otherwise. We prove that the time evolution of a quantum par ..."
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Cited by 21 (1 self)
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We consider Hermitian and symmetric random band matrices H in d � 1 dimensions. The matrix elements Hxy, indexed by x, y ∈ Λ ⊂ Z d, are independent, uniformly distributed random variables if x − y  is less than the band width W, and zero otherwise. We prove that the time evolution of a quantum particle subject to the Hamiltonian H is diffusive on time scales t ≪ W d/3. We also show that the localization length of the eigenvectors of H is larger than a factor W d/6 times the band width. All results are uniform in the size Λ  of the matrix. AMS Subject Classification: 15B52, 82B44, 82C44