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Jun 26, 2018
06/18

by
Olga Klopp

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We consider the matrix completion problem where the aim is to esti-mate a large data matrix for which only a relatively small random subset of its entries is observed. Quite popular approaches to matrix completion problem are iterative thresholding methods. In spite of their empirical success, the theoretical guarantees of such iterative thresholding methods are poorly understood. The goal of this paper is to provide strong theo-retical guarantees, similar to those obtained for nuclear-norm...

Topics: Mathematics, Statistics Theory, Statistics

Source: http://arxiv.org/abs/1502.00146

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38

Sep 24, 2013
09/13

by
Olga Klopp

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We propose a new pivotal method for estimating high-dimensional matrices. Assume that we observe a small set of entries or linear combinations of entries of an unknown matrix $A_0$ corrupted by noise. We propose a new method for estimating $A_0$ which does not rely on the knowledge or an estimation of the standard deviation of the noise $\sigma$. Our estimator achieves, up to a logarithmic factor, optimal rates of convergence under the Frobenius risk and, thus, has the same prediction...

Source: http://arxiv.org/abs/1112.3055v2

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132

Jul 20, 2013
07/13

by
Olga Klopp

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In the present paper we consider the problem of matrix completion with noise for general sampling schemes. Unlike previous works, in our construction we do not need to know or to evaluate the sampling distribution or the variance of the noise. We propose new nuclear-norm penalized estimators, one of them of the "square-root" type. We prove that, up to a logarithmic factor, our estimators achieve optimal rates with respect to the estimation error.

Source: http://arxiv.org/abs/1203.0108v2

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4.0

Jun 30, 2018
06/18

by
Olga Klopp; Nicolas Verzelen

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Consider the twin problems of estimating the connection probability matrix of an inhomogeneous random graph and the graphon of a W-random graph. We establish the minimax estimation rates with respect to the cut metric for classes of block constant matrices and step function graphons. Surprisingly, our results imply that, from the minimax point of view, the raw data, that is, the adjacency matrix of the observed graph, is already optimal and more involved procedures cannot improve the...

Topics: Statistics Theory, Statistics, Mathematics

Source: http://arxiv.org/abs/1703.05101

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49

Sep 18, 2013
09/13

by
Olga Klopp; Marianna Pensky

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In the present paper we consider the varying coefficient model which represents a useful tool for exploring dynamic patterns in many applications. Existing methods typically provide asymptotic evaluation of precision of estimation procedures under the assumption that the number of observations tends to infinity. In practical applications, however, only a finite number of measurements are available. In the present paper we focus on a non-asymptotic approach to the problem. We propose a novel...

Source: http://arxiv.org/abs/1211.3394v2

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7.0

Jun 30, 2018
06/18

by
Alexandra Carpentier; Olga Klopp; Matthias Löffler

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In the present note we consider the problem of constructing honest and adaptive confidence sets for the matrix completion problem. For the Bernoulli model with known variance of the noise we provide a realizable method for constructing confidence sets that adapt to the unknown rank of the true matrix.

Topics: Statistics Theory, Statistics, Mathematics

Source: http://arxiv.org/abs/1704.02760

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4.0

Jun 30, 2018
06/18

by
Olga Klopp; Karim Lounici; Alexandre B. Tsybakov

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This paper considers the problem of recovery of a low-rank matrix in the situation when most of its entries are not observed and a fraction of observed entries are corrupted. The observations are noisy realizations of the sum of a low rank matrix, which we wish to recover, with a second matrix having a complementary sparse structure such as element-wise or column-wise sparsity. We analyze a class of estimators obtained by solving a constrained convex optimization problem that combines the...

Topics: Mathematics, Statistics Theory, Statistics

Source: http://arxiv.org/abs/1412.8132

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9.0

Jun 28, 2018
06/18

by
Olga Klopp; Alexandre B. Tsybakov; Nicolas Verzelen

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Inhomogeneous random graph models encompass many network models such as stochastic block models and latent position models. We consider the problem of statistical estimation of the matrix of connection probabilities based on the observations of the adjacency matrix of the network. Taking the stochastic block model as an approximation, we construct estimators of network connection probabilities -- the ordinary block constant least squares estimator, and its restricted version. We show that they...

Topics: Statistics, Statistics Theory, Mathematics

Source: http://arxiv.org/abs/1507.04118

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3.0

Jun 30, 2018
06/18

by
Jean Lafond; Olga Klopp; Eric Moulines; Jospeh Salmon

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The task of reconstructing a matrix given a sample of observedentries is known as the matrix completion problem. It arises ina wide range of problems, including recommender systems, collaborativefiltering, dimensionality reduction, image processing, quantum physics or multi-class classificationto name a few. Most works have focused on recovering an unknown real-valued low-rankmatrix from randomly sub-sampling its entries.Here, we investigate the case where the observations take a finite number...

Topics: Mathematics, Machine Learning, Statistics Theory, Statistics

Source: http://arxiv.org/abs/1412.2632

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7.0

Jun 29, 2018
06/18

by
Alexandra Carpentier; Olga Klopp; Matthias Löffler; Richard Nickl

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In the present paper we study the problem of existence of honest and adaptive confidence sets for matrix completion. We consider two statistical models: the trace regression model and the Bernoulli model. In the trace regression model, we show that honest confidence sets that adapt to the unknown rank of the matrix exist even when the error variance is unknown. Contrary to this, we prove that in the Bernoulli model, honest and adaptive confidence sets exist only when the error variance is known...

Topics: Statistics, Statistics Theory, Mathematics

Source: http://arxiv.org/abs/1608.04861

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3.0

Jun 30, 2018
06/18

by
Olga Klopp; Jean Lafond; Eric Moulines; Joseph Salmon

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The task of estimating a matrix given a sample of observed entries is known as the \emph{matrix completion problem}. Most works on matrix completion have focused on recovering an unknown real-valued low-rank matrix from a random sample of its entries. Here, we investigate the case of highly quantized observations when the measurements can take only a small number of values. These quantized outputs are generated according to a probability distribution parametrized by the unknown matrix of...

Topics: Mathematics, Machine Learning, Statistics Theory, Statistics

Source: http://arxiv.org/abs/1408.6218