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108
Confidence intervals for lowdimensional parameters with highdimensional data
 ArXiv.org
"... Abstract. The purpose of this paper is to propose methodologies for statistical inference of lowdimensional parameters with highdimensional data. We focus on constructing confidence intervals for individual coefficients and linear combinations of several of them in a linear regression model, alth ..."
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Cited by 28 (1 self)
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Abstract. The purpose of this paper is to propose methodologies for statistical inference of lowdimensional parameters with highdimensional data. We focus on constructing confidence intervals for individual coefficients and linear combinations of several of them in a linear regression model, although our ideas are applicable in a much broader context. The theoretical results presented here provide sufficient conditions for the asymptotic normality of the proposed estimators along with a consistent estimator for their finitedimensional covariance matrices. These sufficient conditions allow the number of variables to far exceed the sample size. The simulation results presented here demonstrate the accuracy of the coverage probability of the proposed confidence intervals, strongly supporting the theoretical results.
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.
Estimation of simultaneously sparse and low rank matrices
 In Proc. ICML
, 2012
"... The paper introduces a penalized matrix estimation procedure aiming at solutions which are sparse and lowrank at the same time. Such structures arise in the context of social networks or protein interactions where underlying graphs have adjacency matrices which are blockdiagonal in the appropria ..."
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Cited by 27 (4 self)
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The paper introduces a penalized matrix estimation procedure aiming at solutions which are sparse and lowrank at the same time. Such structures arise in the context of social networks or protein interactions where underlying graphs have adjacency matrices which are blockdiagonal in the appropriate basis. We introduce a convex mixed penalty which involves `1norm and trace norm simultaneously. We obtain an oracle inequality which indicates how the two effects interact according to the nature of the target matrix. We bound generalization error in the link prediction problem. We also develop proximal descent strategies to solve the optimization problem efficiently and evaluate performance on synthetic and real data sets. 1.
Detection of a sparse submatrix of a highdimensional noisy matrix. Bernoulli,
"... We observe a N × M matrix Y ij = s ij + ξ ij with ξ ij ∼ N (0, 1) i.i.d. in i, j , and s ij ∈ R. We test the null hypothesis s ij = 0 for all i, j against the alternative that there exists some submatrix of size n × m with significant elements in the sense that s ij ≥ a > 0. We propose a test pr ..."
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Cited by 21 (1 self)
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We observe a N × M matrix Y ij = s ij + ξ ij with ξ ij ∼ N (0, 1) i.i.d. in i, j , and s ij ∈ R. We test the null hypothesis s ij = 0 for all i, j against the alternative that there exists some submatrix of size n × m with significant elements in the sense that s ij ≥ a > 0. We propose a test procedure and compute the asymptotical detection boundary a so that the maximal testing risk tends to 0 as We prove that this boundary is asymptotically sharp minimax under some additional constraints. Relations with other testing problems are discussed. We propose a testing procedure which adapts to unknown (n, m) within some given set and compute the adaptive sharp rates. The implementation of our test procedure on synthetic data shows excellent behavior for sparse, not necessarily squared matrices. We extend our sharp minimax results in different directions: first, to Gaussian matrices with unknown variance, next, to matrices of random variables having a distribution from an exponential family (nonGaussian) and, finally, to a twosided alternative for matrices with Gaussian elements.
von Neumann entropy penalization and low rank matrix approximation.
, 2010
"... Abstract We study a problem of estimation of a Hermitian nonnegatively definite matrix ρ of unit trace (for instance, a density matrix of a quantum system) based on n i.i.d. measurements (X 1 , Y 1 ), . . . , (X n , Y n ), where {X j } being random i.i.d. Hermitian matrices and {ξ j } being i.i.d. ..."
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Cited by 19 (2 self)
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Abstract We study a problem of estimation of a Hermitian nonnegatively definite matrix ρ of unit trace (for instance, a density matrix of a quantum system) based on n i.i.d. measurements (X 1 , Y 1 ), . . . , (X n , Y n ), where {X j } being random i.i.d. Hermitian matrices and {ξ j } being i.i.d. random variables with E(ξ j X j ) = 0. The estimator is considered, where S is the set of all nonnegatively definite Hermitian m × m matrices of trace 1. The goal is to derive oracle inequalities showing how the estimation error depends on the accuracy of approximation of the unknown state ρ by lowrank matrices.
Universal lowrank matrix recovery from Pauli measurements
"... We study the problem of reconstructing an unknown matrix M of rank r and dimension d using O(rd poly log d) Pauli measurements. This has applications in quantum state tomography, and is a noncommutative analogue of a wellknown problem in compressed sensing: recovering a sparse vector from a few of ..."
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We study the problem of reconstructing an unknown matrix M of rank r and dimension d using O(rd poly log d) Pauli measurements. This has applications in quantum state tomography, and is a noncommutative analogue of a wellknown problem in compressed sensing: recovering a sparse vector from a few of its Fourier coefficients. We show that almost all sets of O(rd log 6 d) Pauli measurements satisfy the rankr restricted isometry property (RIP). This implies that M can be recovered from a fixed (“universal”) set of Pauli measurements, using nuclearnorm minimization (e.g., the matrix Lasso), with nearlyoptimal bounds on the error. A similar result holds for any class of measurements that use an orthonormal operator basis whose elements have small operator norm. Our proof uses Dudley’s inequality for Gaussian processes, together with bounds on covering numbers obtained via entropy duality. 1
Concentrationbased guarantees for lowrank matrix reconstruction
 24th Annual Conference on Learning Theory (COLT
, 2011
"... We consider the problem of approximately reconstructing a partiallyobserved, approximately lowrank matrix. This problem has received much attention lately, mostly using the tracenorm as a surrogate to the rank. Here we study lowrank matrix reconstruction using both the tracenorm, as well as the ..."
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Cited by 19 (5 self)
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We consider the problem of approximately reconstructing a partiallyobserved, approximately lowrank matrix. This problem has received much attention lately, mostly using the tracenorm as a surrogate to the rank. Here we study lowrank matrix reconstruction using both the tracenorm, as well as the lessstudied maxnorm, and present reconstruction guarantees based on existing analysis on the Rademacher complexity of the unit balls of these norms. We show how these are superior in several ways to recently published guarantees based on specialized analysis.
Blind Deconvolution using Convex Programming
, 2012
"... We consider the problem of recovering two unknown vectors, w and x, of length L from their circular convolution. We make the structural assumption that the two vectors are members known subspaces, one with dimension N and the other with dimension K. Although the observed convolution is nonlinear in ..."
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Cited by 18 (1 self)
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We consider the problem of recovering two unknown vectors, w and x, of length L from their circular convolution. We make the structural assumption that the two vectors are members known subspaces, one with dimension N and the other with dimension K. Although the observed convolution is nonlinear in both w and x, it is linear in the rank1 matrix formed by their outer product wx ∗. This observation allows us to recast the deconvolution problem as lowrank matrix recovery problem from linear measurements, whose natural convex relaxation is a nuclear norm minimization program. We prove the effectiveness of this relaxation by showing that for “generic ” signals, the program can deconvolve w and x exactly when the maximum of N and K is almost on the order of L. That is, we show that if x is drawn from a random subspace of dimension N, and w is a vector in a subspace of dimension K whose basis vectors are “spread out ” in the frequency domain, then nuclear norm minimization recovers wx ∗ without error. We discuss this result in the context of blind channel estimation in communications. If we have a message of length N which we code using a random L × N coding matrix, and the encoded message travels through an unknown linear timeinvariant channel of maximum length K, then the receiver can recover both the channel response and the message when L � N + K, to within constant and log factors. 1