We show that the lower bound on the sum rate of the direct and indirect Gaussian multiterminal source coding problems can be derived in a unified manner by exploiting the semidefinite partial order of the distortion c...
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We show that the lower bound on the sum rate of the direct and indirect Gaussian multiterminal source coding problems can be derived in a unified manner by exploiting the semidefinite partial order of the distortion covariance matrices associated with the minimum mean squared error (MMSE) estimation and the so-called reduced optimal linear estimation, through which an intimate connection between the lower bound and the Berger-Tung upper bound is revealed. We give a new proof of the minimum sum rate of the indirect Gaussian multiterminal source coding problem (i.e., the Gaussian CEO problem). For the direct Gaussian multiterminal source coding problem, we derive a general lower bound on the sum rate and establish a set of sufficient conditions under which the lower bound coincides with the Berger-Tung upper bound. We show that the sufficient conditions are satisfied for a class of sources and distortion constraints;in particular, they hold for arbitrary positive definite source covariance matrices in the high-resolution regime. In contrast with the existing proofs, the new method does not rely on Shannon's entropy power inequality.
We show how to approximate the feasible region of structured convex optimization problems by a family of convex sets with explicitly given and efficient (if the accuracy of the approximation is moderate) self-concorda...
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We show how to approximate the feasible region of structured convex optimization problems by a family of convex sets with explicitly given and efficient (if the accuracy of the approximation is moderate) self-concordant barriers. This approach extends the reach of the modern theory of interior-point methods, and lays the foundation for new ways to treat structured convex optimization problems with a very large number of constraints. Moreover, our approach provides a strong connection from the theory of self-concordant barriers to the combinatorial optimization literature on solving packing and covering problems.
Stability analysis of an aperiodic sampled-data control system is considered for application to networked and embedded control. The stability condition is described in a linear matrix inequality to be satisfied for al...
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Stability analysis of an aperiodic sampled-data control system is considered for application to networked and embedded control. The stability condition is described in a linear matrix inequality to be satisfied for all possible sampling intervals. Although this condition is numerically intractable, a tractable sufficient condition can be constructed with the mean value theorem. Special attention is paid to tightness of the sufficient condition for less conservative stability analysis. A region-dividing technique for the reduction of conservatism and generalization to stabilization are also discussed. An example demonstrates the efficacy of the approach. (C) 2010 Elsevier Ltd. All rights reserved.
We continue the recent line of work on the connection between semidefinite programming (SDP)-based approximation algorithms and the unique games conjecture. Given any Boolean 2-CSP (or, more generally, any Boolean 2-C...
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We continue the recent line of work on the connection between semidefinite programming (SDP)-based approximation algorithms and the unique games conjecture. Given any Boolean 2-CSP (or, more generally, any Boolean 2-CSP with real-valued "predicates"), we show how to reduce the search for a good inapproximability result to a certain numeric minimization problem. Furthermore, we give an SDP-based approximation algorithm and show that the approximation ratio of this algorithm on a certain restricted type of instances is exactly the inapproximability ratio yielded by our hardness result. We conjecture that the restricted type required for the hardness result is in fact no restriction, which would imply that these upper and lower bounds match exactly. This conjecture is supported by all existing results for specific 2-CSPs. As an application, we show that Max 2-AND is unique games-hard to approximate within 0.87435. This improves upon the best previous hardness of alpha(GW) + epsilon approximate to 0.87856 and comes very close to matching the approximation ratio of the best algorithm known, 0.87401. It also establishes that balanced instances of Max 2-AND, i.e., instances in which each variable occurs positively and negatively equally often, are not the hardest to approximate, as these can be approximated within a factor aGW and that Max Cut is not the hardest 2-CSP.
Spectrahedra are sets defined by linear matrix inequalities. Projections of spectrahedra are called semidefinitely representable sets. Both kinds of sets are of practical use in polynomial optimization, since they occ...
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Spectrahedra are sets defined by linear matrix inequalities. Projections of spectrahedra are called semidefinitely representable sets. Both kinds of sets are of practical use in polynomial optimization, since they occur as feasible sets in semidefinite programming. There are several recent results on the question which sets are semidefinitely representable. So far, all results focus on the case of closed sets. In this work we develop a new method to prove semidefinite representability of sets which are not closed. For example, the interior of a semidefinitely representable set is shown to be semidefinitely representable. More general, one can remove faces of a semidefinitely representable set and preserve semidefinite representability, as long as the faces are parametrized in a suitable way. (C) 2010 Elsevier Inc. All rights reserved.
A computational scheme of solving the nonlinear static output feedback design problems for a class of polynomial nonlinear systems is investigated in this paper. Sufficient conditions to achieve the closed-loop stabil...
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A computational scheme of solving the nonlinear static output feedback design problems for a class of polynomial nonlinear systems is investigated in this paper. Sufficient conditions to achieve the closed-loop stability with or without H-infinity performance are presented as state-dependent matrix inequalities, which provides an effective way for the application of the new sum of squares programming technique to obtain computationally tractable solutions. By introducing additional matrix variables, we succeed in eliminating the coupling between system matrices and the Lyapunov matrix. The proposed methodology is also extended to the synthesis for the parameter-dependent polynomial systems. Robust polynomial output feedback controller is designed in an efficient computational manner. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed methodology. Copyright (C) 2009 John Wiley & Sons, Ltd.
A standard and established method for solving a Least Squares problem in the presence of a structured uncertainty is to assemble and solve a semidefinite programming (SDP) equivalent problem. When the problem's di...
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A standard and established method for solving a Least Squares problem in the presence of a structured uncertainty is to assemble and solve a semidefinite programming (SDP) equivalent problem. When the problem's dimensions are high, the solution of the structured robust least squares (RLS) problem via SDP becomes an expensive task in a computational complexity sense. We propose a subgradient based solution that utilizes the MinMax structure of the problem. This algorithm is justified by Danskin's MinMax Theorem and enjoys the well-known convergence properties of the subgradient method. The complexity of the new scheme is analyzed and its efficiency is verified by simulations of a robust equalization design.
In this paper, we study the application of sparse principal component analysis (PCA) to clustering and feature selection problems. Sparse PCA seeks sparse factors, or linear combinations of the data variables, explain...
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In this paper, we study the application of sparse principal component analysis (PCA) to clustering and feature selection problems. Sparse PCA seeks sparse factors, or linear combinations of the data variables, explaining a maximum amount of variance in the data while having only a limited number of nonzero coefficients. PCA is often used as a simple clustering technique and sparse factors allow us here to interpret the clusters in terms of a reduced set of variables. We begin with a brief introduction and motivation on sparse PCA and detail our implementation of the algorithm in d'Aspremont et al. (SIAM Rev. 49(3):434-448, 2007). We then apply these results to some classic clustering and feature selection problems arising in biology.
This technical note tackles the problem of constructing state-feedback stabilizers that guarantee good transient closed-loop performance when applied to general unknown nonlinear multiinput state-feedback stabilizable...
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This technical note tackles the problem of constructing state-feedback stabilizers that guarantee good transient closed-loop performance when applied to general unknown nonlinear multiinput state-feedback stabilizable systems. An adaptive Control Lyapunov Function-based control scheme is proposed in order to address this problem. Mathematical analysis establishes that the proposed control scheme guarantees good closed-loop transient performance outside the regions where the system is uncontrollable, provided the controlled system admits a controllability-like property.
This paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries. This is known as the matrix completion problem, and comes up in a great number of applications, including ...
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This paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries. This is known as the matrix completion problem, and comes up in a great number of applications, including the famous Netflix Prize and other similar questions in collaborative filtering. In general, accurate recovery of a matrix from a small number of entries is impossible, but the knowledge that the unknown matrix has low rank radically changes this premise, making the search for solutions meaningful. This paper presents optimality results quantifying the minimum number of entries needed to recover a matrix of rank r exactly by any method whatsoever (the information theoretic limit). More importantly, the paper shows that, under certain incoherence assumptions on the singular vectors of the matrix, recovery is possible by solving a convenient convex program as soon as the number of entries is on the order of the information theoretic limit (up to logarithmic factors). This convex program simply finds, among all matrices consistent with the observed entries, that with minimum nuclear norm. As an example, we show that on the order of nr log(n) samples are needed to recover a random n x n matrix of rank by any method, and to be sure, nuclear norm minimization succeeds as soon as the number of entries is of the form nr log(n).
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