Almost sure localization of the eigenvalues in a gaussian information plus noise model. Applications to the spiked models

Let $\boldsymbol{\Sigma}_N$ be a $M \times N$ random matrix defined by $\boldsymbol{\Sigma}_N = \mathbf{B}_N + \sigma \mathbf{W}_N$ where $\mathbf{B}_N$ is a uniformly bounded deterministic matrix and where $\mathbf{W}_N$ is an independent identically distributed complex Gaussian matrix with zero mean and variance $\frac{1}{N}$ entries. The purpose of this paper is to study the almost sure location of the eigenvalues $\hat{\lambda}_{1,N} \geq ... \geq \hat{\lambda}_{M,N}$ of the Gram matrix ${\boldsymbol \Sigma}_N {\boldsymbol \Sigma}_N^*$ when $M$ and $N$ converge to $+\infty$ such that the ratio $c_N = \frac{M}{N}$ converges towards a constant $c > 0$. The results are used in order to derive, using an alernative approach, known results concerning the behaviour of the largest eigenvalues of ${\boldsymbol \Sigma}_N {\boldsymbol \Sigma}_N^*$ when the rank of $\mathbf{B}_N$ remains fixed when $M$ and $N$ converge to $+\infty$.Comment: 19 pages, 1 figure, Accepted for publication in Electronic Journal of Probabilit

Improved subspace estimation for multivariate observations of high dimension: the deterministic signals case

We consider the problem of subspace estimation in situations where the number of available snapshots and the observation dimension are comparable in magnitude. In this context, traditional subspace methods tend to fail because the eigenvectors of the sample correlation matrix are heavily biased with respect to the true ones. It has recently been suggested that this situation (where the sample size is small compared to the observation dimension) can be very accurately modeled by considering the asymptotic regime where the observation dimension $M$ and the number of snapshots $N$ converge to $+\infty$ at the same rate. Using large random matrix theory results, it can be shown that traditional subspace estimates are not consistent in this asymptotic regime. Furthermore, new consistent subspace estimate can be proposed, which outperform the standard subspace methods for realistic values of $M$ and $N$. The work carried out so far in this area has always been based on the assumption that the observations are random, independent and identically distributed in the time domain. The goal of this paper is to propose new consistent subspace estimators for the case where the source signals are modelled as unknown deterministic signals. In practice, this allows to use the proposed approach regardless of the statistical properties of the source signals. In order to construct the proposed estimators, new technical results concerning the almost sure location of the eigenvalues of sample covariance matrices of Information plus Noise complex Gaussian models are established. These results are believed to be of independent interest.Comment: New version with minor corrections. The present paper is an extended version of a paper (same title) to appear in IEEE Trans. on Information Theor

Continuous dependence estimate for a degenerate parabolic-hyperbolic equation with Levy noise

In this article, we are concerned with a multidimensional degenerate parabolic-hyperbolic equation driven by Levy processes. Using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities of the entropy solutions under the assumption that Levy noise depends only on the solution. This result is used to show the error estimate for the stochastic vanishing viscosity method. In addition, we establish fractional BV estimate for vanishing viscosity approximations in case the noise coefficients depend on both the solution and spatial variable.Comment: 31 Pages. arXiv admin note: text overlap with arXiv:1502.0249

Implicit search trails for video recommendation

In this demo paper we demonstrate our approach and system for using implicit actions involved in video search to provide recommendations to users. The goal of this system is to improve the quality of the results that users find, and in doing so, help users to explore a large and difficult information space and help them consider search options that they may not have considered otherwise. Results of a user evaluation show that this approach achieves all of these goals

Search trails using user feedback to improve video search

In this paper we present an innovative approach for aiding users in the difficult task of video search. We use community based feedback mined from the interactions of previous users of our video search system to aid users in their search tasks. This feedback is the basis for providing recommendations to users of our video retrieval system. The ultimate goal of this system is to improve the quality of the results that users find, and in doing so, help users to explore a large and difficult information space and help them consider search options that they may not have considered otherwise. In particular we wish to make the difficult task of search for video much easier for users. The results of a user evaluation indicate that we achieved our goals, the performance of the users in retrieving relevant videos improved, and users were able to explore the collection to a greater extent