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The mathematics of eigenvalue optimization
 MATHEMATICAL PROGRAMMING
"... Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of classical mathematical techniques and contemp ..."
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Cited by 115 (11 self)
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Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of classical mathematical techniques
Distance metric learning with eigenvalue optimization
 Journal of Machine Learning Research (Special Topics on Kernel and Metric Learning
, 2012
"... The main theme of this paper is to develop a novel eigenvalue optimization framework for learning a Mahalanobis metric. Within this context, we introduce a novel metric learning approach called DMLeig which is shown to be equivalent to a wellknown eigenvalue optimization problem called minimizing ..."
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Cited by 46 (2 self)
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The main theme of this paper is to develop a novel eigenvalue optimization framework for learning a Mahalanobis metric. Within this context, we introduce a novel metric learning approach called DMLeig which is shown to be equivalent to a wellknown eigenvalue optimization problem called minimizing
ON EIGENVALUE OPTIMIZATION* ALEXANDER SHAPIRO
"... Abstract. In this paper we study optimization problems involving eigenvalues of symmetric matrices. One of the difficulties with numerical analysis of such problems is that the eigenvalues, considered as functions of a symmetric matrix, are not differentiable at those points where they coalesce. We ..."
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Abstract. In this paper we study optimization problems involving eigenvalues of symmetric matrices. One of the difficulties with numerical analysis of such problems is that the eigenvalues, considered as functions of a symmetric matrix, are not differentiable at those points where they coalesce. We
On the eigenvalues optimization of beams with damping patches
"... Abstract: The paper discusses the behavior of beams with external nonlocal damping patches made from traditional and auxetic materials. Unlike ordinary local damping models, the nonlocal damping force is modeled as a weighted average of the velocity field over the spatial domain, determined by a ke ..."
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Cited by 1 (0 self)
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kernel function based on distance measures. The performance with respect to eigenvalues is discussed in order to avoid resonance. The optimization is performed by determining the location of patches from maximizing eigenvalues or gap between them. KeyWords: eigenvalues, optimization, damping patches
Column Subset Selection, Matrix Factorization, and Eigenvalue Optimization
, 2008
"... Given a fixed matrix, the problem of column subset selection requests a column submatrix that has favorable spectral properties. Most research from the algorithms and numerical linear algebra communities focuses on a variant called rankrevealing QR, which seeks a wellconditioned collection of colu ..."
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Cited by 20 (1 self)
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Given a fixed matrix, the problem of column subset selection requests a column submatrix that has favorable spectral properties. Most research from the algorithms and numerical linear algebra communities focuses on a variant called rankrevealing QR, which seeks a wellconditioned collection of columns that spans the (numerical) range of the matrix. The functional analysis literature contains another strand of work on column selection whose algorithmic implications have not been explored. In particular, a celebrated result of Bourgain and Tzafriri demonstrates that each matrix with normalized columns contains a large column submatrix that is exceptionally well conditioned. Unfortunately, standard proofs of this result cannot be regarded as algorithmic. This paper presents
Eigenvalue Optimization in C 2 Subdivision and Boundary Subdivision
, 2011
"... die mir täglich die nötige Freude und Energie gegeben haben. ..."
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Cited by 3 (1 self)
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die mir täglich die nötige Freude und Energie gegeben haben.
Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming
 Journal of the ACM
, 1995
"... We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution ..."
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Cited by 1211 (13 self)
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We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds
Randomized Gossip Algorithms
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 2006
"... Motivated by applications to sensor, peertopeer, and ad hoc networks, we study distributed algorithms, also known as gossip algorithms, for exchanging information and for computing in an arbitrarily connected network of nodes. The topology of such networks changes continuously as new nodes join a ..."
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Cited by 532 (5 self)
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method that solves the optimization problem over the network. The relation of averaging time to the second largest eigenvalue naturally relates it to the mixing time of a random walk with transition probabilities derived from the gossip algorithm. We use this connection to study the performance
Laplacian eigenmaps and spectral techniques for embedding and clustering.
 Proceeding of Neural Information Processing Systems,
, 2001
"... Abstract Drawing on the correspondence between the graph Laplacian, the LaplaceBeltrami op erator on a manifold , and the connections to the heat equation , we propose a geometrically motivated algorithm for constructing a representation for data sampled from a low dimensional manifold embedded in ..."
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Cited by 668 (7 self)
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of the manifold on which the data may possibly reside. Recently, there has been some interest (Tenenbaum et aI, 2000 ; The core algorithm is very simple, has a few local computations and one sparse eigenvalu e problem. The solution reflects th e intrinsic geom etric structure of the manifold. Th e justification
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