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Estimation of (near) lowrank matrices with noise and highdimensional scaling
"... We study an instance of highdimensional statistical inference in which the goal is to use N noisy observations to estimate a matrix Θ ∗ ∈ R k×p that is assumed to be either exactly low rank, or “near ” lowrank, meaning that it can be wellapproximated by a matrix with low rank. We consider an Me ..."
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Cited by 93 (14 self)
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We study an instance of highdimensional statistical inference in which the goal is to use N noisy observations to estimate a matrix Θ ∗ ∈ R k×p that is assumed to be either exactly low rank, or “near ” lowrank, meaning that it can be wellapproximated by a matrix with low rank. We consider an Mestimator based on regularization by the traceornuclearnormovermatrices, andanalyze its performance under highdimensional scaling. We provide nonasymptotic bounds on the Frobenius norm error that hold for a generalclassofnoisyobservationmodels,and apply to both exactly lowrank and approximately lowrank matrices. We then illustrate their consequences for a number of specific learning models, including lowrank multivariate or multitask regression, system identification in vector autoregressive processes, and recovery of lowrank matrices from random projections. Simulations show excellent agreement with the highdimensional scaling of the error predicted by our theory. 1.
Nonconcave penalized likelihood with NPdimensionality
 IEEE Trans. Inform. Theor
, 2011
"... Abstract—Penalized likelihood methods are fundamental to ultrahigh dimensional variable selection. How high dimensionality such methods can handle remains largely unknown. In this paper, we show that in the context of generalized linear models, such methods possess model selection consistency with o ..."
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Cited by 51 (14 self)
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Abstract—Penalized likelihood methods are fundamental to ultrahigh dimensional variable selection. How high dimensionality such methods can handle remains largely unknown. In this paper, we show that in the context of generalized linear models, such methods possess model selection consistency with oracle properties even for dimensionality of nonpolynomial (NP) order of sample size, for a class of penalized likelihood approaches using foldedconcave penalty functions, which were introduced to ameliorate the bias problems of convex penalty functions. This fills a longstanding gap in the literature where the dimensionality is allowed to grow slowly with the sample size. Our results are also applicable to penalized likelihood with the L1penalty, which is a convex function at the boundary of the class of foldedconcave penalty functions under consideration. The coordinate optimization is implemented for finding the solution paths, whose performance is evaluated by a few simulation examples and the real data analysis. Index Terms—Coordinate optimization, foldedconcave penalty, high dimensionality, Lasso, nonconcave penalized likelihood, oracle property, SCAD, variable selection, weak oracle property. I.
2011): “Variance estimation using refitted crossvalidation in ultrahigh dimensional regression,” forthcoming
 INFERENCE AFTER MODEL SELECTION 59
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Limiting laws of coherence of random matrices with applications to testing covariance structure and construction of compressed sensing matrices
 Ann. Stat
, 2011
"... Testing covariance structure is of significant interest in many areas of statistical analysis and construction of compressed sensing matrices is an important problem in signal processing. Motivated by these applications, we study in this paper the limiting laws of the coherence of an n×p random matr ..."
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Cited by 29 (9 self)
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Testing covariance structure is of significant interest in many areas of statistical analysis and construction of compressed sensing matrices is an important problem in signal processing. Motivated by these applications, we study in this paper the limiting laws of the coherence of an n×p random matrix in the highdimensional setting where p can be much larger than n. Both the law of large numbers and the limiting distribution are derived. We then consider testing the bandedness of the covariance matrix of a high dimensional Gaussian distribution which includes testing for independence as a special case. The limiting laws of the coherence of the data matrix play a critical role in the construction of the test. We also apply the asymptotic results to the construction of compressed sensing matrices.
Phase transition in limiting distributions of coherence of highdimensional random matrices
, 2012
"... The coherence of a random matrix, which is defined to be the largest magnitude of the Pearson correlation coefficients between the columns of the random matrix, is an important quantity for a wide range of applications including highdimensional statistics and signal processing. Inspired by these ap ..."
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Cited by 19 (5 self)
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The coherence of a random matrix, which is defined to be the largest magnitude of the Pearson correlation coefficients between the columns of the random matrix, is an important quantity for a wide range of applications including highdimensional statistics and signal processing. Inspired by these applications, this paper studies the limiting laws of the coherence of n × p random matrices for a full range of the dimension p with a special focus on the ultra highdimensional setting. Assuming the columns of the random matrix are independent random vectors with a common spherical distribution, we give a complete characterization of the behavior of the limiting distributions of the coherence. More specifically, the limiting distributions of the coherence are derived log p → ∞. The results show that the limiting behavior of the coherence differs significantly in different regimes and exhibits interesting phase transition phenomena as the dimension p grows as a function of n. Applications to statistics and compressed sensing in the ultra highdimensional setting are also discussed. separately for three regimes: 1 n log p → 0,
Covariance Estimation: The GLM and Regularization Perspectives
"... Finding an unconstrained and statistically interpretable reparameterization of a covariance matrix is still an open problem in statistics. Its solution is of central importance in covariance estimation, particularly in the recent highdimensional data environment where enforcing the positivedefinit ..."
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Cited by 14 (2 self)
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Finding an unconstrained and statistically interpretable reparameterization of a covariance matrix is still an open problem in statistics. Its solution is of central importance in covariance estimation, particularly in the recent highdimensional data environment where enforcing the positivedefiniteness constraint could be computationally expensive. We provide a survey of the progress made in modeling covariance matrices from the perspectives of generalized linear models (GLM) or parsimony and use of covariates in low dimensions, regularization (shrinkage, sparsity) for highdimensional data, and the role of various matrix factorizations. A viable and emerging regressionbased setup which is suitable for both the GLM and the regularization approaches is to link a covariance matrix, its inverse or their factors to certain regression models and then solve the relevant (penalized) least squares problems. We point out several instances of this regressionbased setup in the literature. A notable case is in the Gaussian graphical models where linear regressions with LASSO penalty are used to estimate the neighborhood of one node at a time (Meinshausen and Bühlmann, 2006). Some advantages
Variable selection with error control: another look at stability selection
, 2011
"... Summary. Stability selection was recently introduced by Meinshausen and Bühlmann as a very general technique designed to improve the performance of a variable selection algorithm. It is based on aggregating the results of applying a selection procedure to subsamples of the data. We introduce a varia ..."
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Cited by 13 (3 self)
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Summary. Stability selection was recently introduced by Meinshausen and Bühlmann as a very general technique designed to improve the performance of a variable selection algorithm. It is based on aggregating the results of applying a selection procedure to subsamples of the data. We introduce a variant, called complementary pairs stability selection, and derive bounds both on the expected number of variables included by complementary pairs stability selection that have low selection probability under the original procedure, and on the expected number of high selection probability variables that are excluded. These results require no (e.g. exchangeability) assumptions on the underlying model or on the quality of the original selection procedure. Under reasonable shape restrictions, the bounds can be further tightened, yielding improved error control, and therefore increasing the applicability of the methodology.
Sparse methods for biomedical data
 SIGKDD Explor. Newsl
, 2012
"... Following recent technological revolutions, the investigation of massive biomedical data with growing scale, diversity, and complexity has taken a center stage in modern data analysis. Although complex, the underlying representations of many biomedical data are often sparse. For example, for a certa ..."
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Cited by 11 (2 self)
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Following recent technological revolutions, the investigation of massive biomedical data with growing scale, diversity, and complexity has taken a center stage in modern data analysis. Although complex, the underlying representations of many biomedical data are often sparse. For example, for a certain disease such as leukemia, even though humans have tens of thousands of genes, only a few genes are relevant to the disease; a gene network is sparse since a regulatory pathway involves only a small number of genes; many biomedical signals are sparse or compressible in the sense that they have concise representations when expressed in a proper basis. Therefore, finding sparse representations is fundamentally important for scientific discovery. Sparse methods based on the ℓ1 norm have attracted a great amount of research efforts in the past decade due to its sparsityinducing property, convenient convexity, and strong theoretical guarantees. They have achieved great success in various applications such as biomarker selection, biological network construction, and magnetic resonance imaging. In this paper, we review stateoftheart sparse methods and their applications to biomedical data.
Sharp Support Recovery from Noisy Random Measurements by ℓ1 minimization
, 2011
"... In this paper, we investigate the theoretical guarantees of penalized ℓ1minimization (also called Basis Pursuit Denoising or Lasso) in terms of sparsity pattern recovery (support and sign consistency) from noisy measurements with nonnecessarily random noise, when the sensing operator belongs to th ..."
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Cited by 11 (5 self)
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In this paper, we investigate the theoretical guarantees of penalized ℓ1minimization (also called Basis Pursuit Denoising or Lasso) in terms of sparsity pattern recovery (support and sign consistency) from noisy measurements with nonnecessarily random noise, when the sensing operator belongs to the Gaussian ensemble (i.e. random design matrix with i.i.d. Gaussian entries). More precisely, we derive sharp nonasymptotic bounds on the sparsity level and (minimal) signaltonoise ratio that ensure support identification for most signals and most Gaussian sensing matrices by solving the Lasso with an appropriately chosen regularization parameter. Our first purpose is to establish conditions allowing exact sparsity pattern recovery when the signal is strictly sparse. Then, these conditions are extended to cover the compressible or nearly sparse case. In these two results, the role of the minimal signaltonoise ratio is crucial. Our third main result gets rid of this assumption in the strictly sparse case, but this time, the Lasso allows only partial recovery of the support. We also provide in this case a sharp ℓ2consistency result on the coefficient vector. The results of the present work have several distinctive features compared to previous ones. One of them is that the leading constants involved in all the bounds are sharp and explicit. This is illustrated by some numerical experiments where it is indeed shown that the sharp sparsity level threshold identified by our theoretical results below which sparsistency of the Lasso solution is guaranteed meets the one empirically observed.