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227
Matrix estimation by universal singular value thresholding
, 2012
"... Abstract. Consider the problem of estimating the entries of a large matrix, when the observed entries are noisy versions of a small random fraction of the original entries. This problem has received widespread attention in recent times, especially after the pioneering works of Emmanuel Candès and ..."
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Abstract. Consider the problem of estimating the entries of a large matrix, when the observed entries are noisy versions of a small random fraction of the original entries. This problem has received widespread attention in recent times, especially after the pioneering works of Emmanuel Candès and collaborators. This paper introduces a simple estimation procedure, called Universal Singular Value Thresholding (USVT), that works for any matrix that has ‘a little bit of structure’. Surprisingly, this simple estimator achieves the minimax error rate up to a constant factor. The method is applied to solve problems related to low rank matrix estimation, blockmodels, distance matrix completion, latent space models, positive definite matrix completion, graphon estimation, and generalized Bradley–Terry models for pairwise comparison. 1.
Universality in Polytope Phase Transitions and Message Passing Algorithms
, 2012
"... We consider a class of nonlinear mappings FA,N in R N indexed by symmetric random matrices A ∈ R N×N with independent entries. Within spin glass theory, special cases of these mappings correspond to iterating the TAP equations and were studied by Erwin Bolthausen. Within information theory, they are ..."
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Cited by 24 (4 self)
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We consider a class of nonlinear mappings FA,N in R N indexed by symmetric random matrices A ∈ R N×N with independent entries. Within spin glass theory, special cases of these mappings correspond to iterating the TAP equations and were studied by Erwin Bolthausen. Within information theory, they are known as ‘approximate message passing ’ algorithms. We study the highdimensional (large N) behavior of the iterates of F for polynomial functions F, and prove that it is universal, i.e. it depends only on the first two moments of the entries of A, under a subgaussian tail condition. As an application, we prove the universality of a certain phase transition arising in polytope geometry and compressed sensing. This solves –for a broad class of random projections – a conjecture by David Donoho and Jared Tanner. 1 Introduction and main results Let A ∈ RN×N be a random Wigner matrix, i.e. a random matrix with i.i.d. entries Aij satisfying E{Aij} = 0 and E{A2 ij} = 1/N. Considerable effort has been devoted to studying the distribution of the eigenvalues of such a matrix [AGZ09, BS05, TV12]. The universality phenomenon is a striking recurring theme in these studies. Roughly speaking, many asymptotic properties of the joint eigenvalues
The single ring theorem
, 2009
"... We study the empirical measure LAn of the eigenvalues of nonnormal square matrices of the form An = UnDnVn with Un,Vn independent Haar distributed on the unitary group and Dn real diagonal. We show that when the empirical measure of the eigenvalues of Dn converges, and Dn satisfies some technical c ..."
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We study the empirical measure LAn of the eigenvalues of nonnormal square matrices of the form An = UnDnVn with Un,Vn independent Haar distributed on the unitary group and Dn real diagonal. We show that when the empirical measure of the eigenvalues of Dn converges, and Dn satisfies some technical conditions, LAn converges towards a rotationally invariant measure on the complex plane whose support is a single ring. In particular, we provide a complete proof of FeinbergZee single ring theorem [5]. We also consider the case where Un,Vn are independent Haar distributed on the orthogonal group. 1 The problem Horn [15] asked the question of describing the eigenvalues of a square matrix with prescribed singular values. If A is a n × n matrix with singular values s1 ≥... ≥
Asymptotic Expansion of β Matrix Models in the Onecut Regime
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2012
"... We prove the existence of a 1/N expansion to all orders in β matrix models with a confining, offcritical potential corresponding to an equilibrium measure with a connected support. Thus, the coefficients of the expansion can be obtained recursively by the “topological recursion ” derived in Chekhov ..."
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Cited by 23 (3 self)
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We prove the existence of a 1/N expansion to all orders in β matrix models with a confining, offcritical potential corresponding to an equilibrium measure with a connected support. Thus, the coefficients of the expansion can be obtained recursively by the “topological recursion ” derived in Chekhov and Eynard (JHEP 0612:026, 2006). Our method relies on the combination of a priori bounds on the correlators and the study of SchwingerDyson equations, thanks to the uses of classical complex analysis techniques. These a priori bounds can be derived following (Boutet de Monvel et al. in J Stat Phys 79(3–4):585–611, 1995; Johansson in Duke Math J 91(1):151–204, 1998; Kriecherbauer and Shcherbina in Fluctuations of eigenvalues of matrix models and their applications, 2010) or for strictly convex potentials by using concentration of measure (Anderson et al. in An introduction to random matrices, Sect. 2.3, Cambridge University Press, Cambridge, 2010). Doing so, we extend the strategy of Guionnet and MaurelSegala (Ann Probab 35:2160–2212, 2007), from the hermitian models (β = 2) and perturbative potentials, to general β models. The existence of the first correction in 1/N was considered in Johansson (1998) and more recently in Kriecherbauer and Shcherbina (2010). Here, by taking similar hypotheses, we extend the result to all orders in 1/N.
Eigenvector distribution of Wigner matrices
"... We consider N ×N Hermitian or symmetric random matrices with independent entries. The distribution of the (i, j)th matrix element is given by a probability measure νij whose first two moments coincide with those of the corresponding Gaussian ensemble. We prove that the joint probability distributio ..."
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We consider N ×N Hermitian or symmetric random matrices with independent entries. The distribution of the (i, j)th matrix element is given by a probability measure νij whose first two moments coincide with those of the corresponding Gaussian ensemble. We prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding Gaussian ensemble. For eigenvectors associated with bulk eigenvalues, the same conclusion holds provided the first four moments of the distribution νij coincide with those of the corresponding Gaussian ensemble. More generally, we prove that the joint eigenvectoreigenvalue distributions near the spectral edge of two generalized Wigner ensembles agree, provided that the first two moments of the entries match and that one of the ensembles satisfies a level repulsion estimate. If in addition the first four moments match then this result holds also in the bulk.
Universality of general βensembles
, 2011
"... We prove the universality of the βensembles with convex analytic potentials and for any β> 0, i.e. we show that the spacing distributions of loggases at any inverse temperature β coincide with those of the Gaussian βensembles. ..."
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Cited by 22 (4 self)
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We prove the universality of the βensembles with convex analytic potentials and for any β> 0, i.e. we show that the spacing distributions of loggases at any inverse temperature β coincide with those of the Gaussian βensembles.
A.: Quantum diffusion and delocalization for band matrices with general distribution
"... 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 and their variances satisfy σ 2 xy: = EHxy  2 = W −d f((x − y)/W) for some probability density f. We assume that the law of each matrix element Hxy is ..."
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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 and their variances satisfy σ 2 xy: = EHxy  2 = W −d f((x − y)/W) for some probability density f. We assume that the law of each matrix element Hxy is symmetric and exhibits subexponential decay. 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 W. All results are uniform in the size Λ  of the matrix. This extends our recent result [1] to general band matrices. As another consequence of our proof we show that, for a larger class of random matrices satisfying x σ2 xy = 1 for all y, the largest eigenvalue of H is bounded with high probability by 2 + M −2/3+ε for any ε> 0, where M: = 1/(maxx,y σ 2 xy).
Bulk Universality of General βEnsembles with Nonconvex Potential
, 2012
"... We prove the bulk universality of the βensembles with nonconvex regular analytic potentials for any β> 0. This removes the convexity assumption appeared in the earlier work [6]. The convexity condition enabled us to use the logarithmic Sobolev inequality to estimate events with small probabilit ..."
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We prove the bulk universality of the βensembles with nonconvex regular analytic potentials for any β> 0. This removes the convexity assumption appeared in the earlier work [6]. The convexity condition enabled us to use the logarithmic Sobolev inequality to estimate events with small probability. The new idea is to introduce a “convexified measure ” so that the local statistics are preserved under this convexification.
Limits of spiked random matrices
, 2013
"... Given a large, highdimensional sample from a spiked population, the top sample covariance eigenvalue is known to exhibit a phase transition. We show that the largest eigenvalues have asymptotic distributions near the phase transition in the rank one spiked real Wishart setting and its general β ana ..."
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Cited by 17 (2 self)
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Given a large, highdimensional sample from a spiked population, the top sample covariance eigenvalue is known to exhibit a phase transition. We show that the largest eigenvalues have asymptotic distributions near the phase transition in the rank one spiked real Wishart setting and its general β analogue, proving a conjecture of Baik, Ben Arous and Péche ́ (2005). We also treat shifted mean Gaussian orthogonal and β ensembles. Such results are entirely new in the real case; in the complex case we strengthen existing results by providing optimal scaling assumptions. One obtains the known limiting random Schrödinger operator on the halfline, but the boundary condition now depends on the perturbation. We derive several characterizations of the limit laws in which β appears as a parameter, including a simple linear boundary value problem. This PDE description recovers known explicit formulas at β = 2, 4, yielding in particular a new and simple proof of the Painleve ́ representations for these