Fast computation of some asymptotic functional inverses

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Decay properties of spectral projectors with applications to electronic structure

Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our theory leads to a rigorous proof of the exponential off-diagonal decay ("nearsightedness") for the density matrix of gapped systems at zero electronic temperature in both orthogonal and non-orthogonal representations, thus providing a firm theoretical basis for the possibility of linear scaling methods in electronic structure calculations for non-metallic systems. We further discuss the case of density matrices for metallic systems at positive electronic temperature. A few other possible applications are also discussed.Comment: 63 pages, 13 figure

Nonlinear Markov Processes in Big Networks

Big networks express various large-scale networks in many practical areas such as computer networks, internet of things, cloud computation, manufacturing systems, transportation networks, and healthcare systems. This paper analyzes such big networks, and applies the mean-field theory and the nonlinear Markov processes to set up a broad class of nonlinear continuous-time block-structured Markov processes, which can be applied to deal with many practical stochastic systems. Firstly, a nonlinear Markov process is derived from a large number of interacting big networks with symmetric interactions, each of which is described as a continuous-time block-structured Markov process. Secondly, some effective algorithms are given for computing the fixed points of the nonlinear Markov process by means of the UL-type RG-factorization. Finally, the Birkhoff center, the Lyapunov functions and the relative entropy are used to analyze stability or metastability of the big network, and several interesting open problems are proposed with detailed interpretation. We believe that the results given in this paper can be useful and effective in the study of big networks.Comment: 28 pages in Special Matrices; 201

Adaptive estimation of covariance matrices via Cholesky decomposition

This paper studies the estimation of a large covariance matrix. We introduce a novel procedure called ChoSelect based on the Cholesky factor of the inverse covariance. This method uses a dimension reduction strategy by selecting the pattern of zero of the Cholesky factor. Alternatively, ChoSelect can be interpreted as a graph estimation procedure for directed Gaussian graphical models. Our approach is particularly relevant when the variables under study have a natural ordering (e.g. time series) or more generally when the Cholesky factor is approximately sparse. ChoSelect achieves non-asymptotic oracle inequalities with respect to the Kullback-Leibler entropy. Moreover, it satisfies various adaptive properties from a minimax point of view. We also introduce and study a two-stage procedure that combines ChoSelect with the Lasso. This last method enables the practitioner to choose his own trade-off between statistical efficiency and computational complexity. Moreover, it is consistent under weaker assumptions than the Lasso. The practical performances of the different procedures are assessed on numerical examples

Oversampling in shift-invariant spaces with a rational sampling period

8 pages, no figures.It is well known that, under appropriate hypotheses, a sampling formula allows us to recover any function in a principal shift-invariant space from its samples taken with sampling period one. Whenever the generator of the shift-invariant space satisfies the Strang-Fix conditions of order r, this formula also provides an approximation scheme of order r valid for smooth functions. In this paper we obtain sampling formulas sharing the same features by using a rational sampling period less than one. With the use of this oversampling technique, there is not one but an infinite number of sampling formulas. Whenever the generator has compact support, among these formulas it is possible to find one whose associated reconstruction functions have also compact support.This work has been supported by the Grant MTM2009-08345 from the D.G.I. of the Spanish Ministerio de Ciencia y Tecnología