Noisy low-rank matrix completion with general sampling distribution

In the present paper, we consider the problem of matrix completion with noise. Unlike previous works, we consider quite general sampling distribution and we do not need to know or to estimate the variance of the noise. Two new nuclear-norm penalized estimators are proposed, one of them of "square-root" type. We analyse their performance under high-dimensional scaling and provide non-asymptotic bounds on the Frobenius norm error. Up to a logarithmic factor, these performance guarantees are minimax optimal in a number of circumstances.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ486 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Validation of Matching

We introduce a technique to compute probably approximately correct (PAC) bounds on precision and recall for matching algorithms. The bounds require some verified matches, but those matches may be used to develop the algorithms. The bounds can be applied to network reconciliation or entity resolution algorithms, which identify nodes in different networks or values in a data set that correspond to the same entity. For network reconciliation, the bounds do not require knowledge of the network generation process

Concentration inequalities for sampling without replacement

Concentration inequalities quantify the deviation of a random variable from a fixed value. In spite of numerous applications, such as opinion surveys or ecological counting procedures, few concentration results are known for the setting of sampling without replacement from a finite population. Until now, the best general concentration inequality has been a Hoeffding inequality due to Serfling [Ann. Statist. 2 (1974) 39-48]. In this paper, we first improve on the fundamental result of Serfling [Ann. Statist. 2 (1974) 39-48], and further extend it to obtain a Bernstein concentration bound for sampling without replacement. We then derive an empirical version of our bound that does not require the variance to be known to the user.Comment: Published at http://dx.doi.org/10.3150/14-BEJ605 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Von Neumann Entropy Penalization and Low Rank Matrix Estimation

A problem of statistical estimation of a Hermitian nonnegatively definite matrix of unit trace (for instance, a density matrix in quantum state tomography) is studied. The approach is based on penalized least squares method with a complexity penalty defined in terms of von Neumann entropy. A number of oracle inequalities have been proved showing how the error of the estimator depends on the rank and other characteristics of the oracles. The methods of proofs are based on empirical processes theory and probabilistic inequalities for random matrices, in particular, noncommutative versions of Bernstein inequality