Approximating the ground state of fermion system by multiple determinant states: matching pursuit approach

We present a simple and stable numerical method to approximate the ground state of a quantum many-body system by multiple determinant states. This method searches these determinant states one by one according to the matching pursuit algorithm. The first determinant state is identical to that of the Hartree-Fock theory. Calculations for two-dimensional Hubbard model serve as a demonstration.Comment: 5 Pages, 1 figur

AN L1 CRITERION FOR DICTIONARY LEARNING BY SUBSPACE IDENTIFICATION

Future and Emerging Technologies (FET) programme within the Seventh Framework Programme for Research of the European Commission, under FET-Open grant number: 225913 (project SMALL).EPSRC Leadership Fellowship (EP/G007177/1

Exact Recovery Conditions for Sparse Representations with Partial Support Information

We address the exact recovery of a k-sparse vector in the noiseless setting when some partial information on the support is available. This partial information takes the form of either a subset of the true support or an approximate subset including wrong atoms as well. We derive a new sufficient and worst-case necessary (in some sense) condition for the success of some procedures based on lp-relaxation, Orthogonal Matching Pursuit (OMP) and Orthogonal Least Squares (OLS). Our result is based on the coherence "mu" of the dictionary and relaxes the well-known condition mu<1/(2k-1) ensuring the recovery of any k-sparse vector in the non-informed setup. It reads mu<1/(2k-g+b-1) when the informed support is composed of g good atoms and b wrong atoms. We emphasize that our condition is complementary to some restricted-isometry based conditions by showing that none of them implies the other. Because this mutual coherence condition is common to all procedures, we carry out a finer analysis based on the Null Space Property (NSP) and the Exact Recovery Condition (ERC). Connections are established regarding the characterization of lp-relaxation procedures and OMP in the informed setup. First, we emphasize that the truncated NSP enjoys an ordering property when p is decreased. Second, the partial ERC for OMP (ERC-OMP) implies in turn the truncated NSP for the informed l1 problem, and the truncated NSP for p<1.Comment: arXiv admin note: substantial text overlap with arXiv:1211.728

Highly sparse representations from dictionaries are unique and independent of the sparseness measure

AbstractThe purpose of this paper is to study sparse representations of signals from a general dictionary in a Banach space. For so-called localized frames in Hilbert spaces, the canonical frame coefficients are shown to provide a near sparsest expansion for several sparseness measures. However, for frames which are not localized, this no longer holds true and sparse representations may depend strongly on the choice of the sparseness measure. A large class of admissible sparseness measures is introduced, and we give sufficient conditions for having a unique sparse representation of a signal from the dictionary w.r.t. such a sparseness measure. Moreover, we give sufficient conditions on a signal such that the simple solution of a linear programming problem simultaneously solves all the nonconvex (and generally hard combinatorial) problems of sparsest representation of the signal w.r.t. arbitrary admissible sparseness measures

On approximation with spline generated framelets

International audienceWe characterize the approximation spaces associated with the best $n$-term approximation in $L_p(\mathbb{R})$ by elements from a tight wavelet frame associated with a spline scaling function. The approximation spaces are shown to be interpolation spaces between $L_p$ and classical Besov spaces, and the result coincide with the result for nonlinear approximation with an orthonormal wavelet with the same smoothness as the spline scaling function. We also show that, under certain conditions, the Besov smoothness can be measured in terms of the sparsity of expansions in the wavelet frame, just like the nonredundant wavelet case. However the characterization now holds even for wavelet frame systems that do not have the usually required number of vanishing moments, {\em e.g.}\ for systems built through the Unitary Extension Principle, which can have no more than one vanishing moment. Using these results, we describe a fast algorithm that takes as input any function and provides a near sparsest expansion of it in the framelet system as well as approximants that reach the optimal rate of nonlinear approximation. Together with the existence of a fast algorithm, the absence of need for vanishing moments may have an important qualitative impact for applications to signal compression, as high vanishing moments usually introduce Gibbs-type phenomenon (or ``ringing'' artifacts) in the approximants

A CONSTRAINED MATCHING PURSUIT APPROACH TO AUDIO DECLIPPING

© 2011 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works

Audio Inpainting

(c) 2012 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works. Published version: IEEE Transactions on Audio, Speech and Language Processing 20(3): 922-932, Mar 2012. DOI: 10.1090/TASL.2011.2168211

Group polytope faces pursuit for recovery of block-sparse signals

This is the accepted version of the article. The final publication is available at link.springer.com. http://www.springerlink.com/content/e0r61416446277w0

A Multichannel Spatial Compressed Sensing Approach for Direction of Arrival Estimation

The final publication is available at http://link.springer.com/chapter/10.1007%2F978-3-642-15995-4_57ESPRC Leadership Fellowship EP/G007144/1EPSRC Platform Grant EP/045235/1EU FET-Open Project FP7-ICT-225913\"SMALL