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833
Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems
 Journal of the ACM
, 1998
"... Abstract. We present a polynomial time approximation scheme for Euclidean TSP in fixed dimensions. For every fixed c Ͼ 1 and given any n nodes in 2 , a randomized version of the scheme finds a (1 ϩ 1/c)approximation to the optimum traveling salesman tour in O(n(log n) O(c) ) time. When the nodes ..."
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Cited by 397 (2 self)
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using any geometric norm (such as ᐉ p for p Ն 1 or other Minkowski norms). They also have simple parallel (i.e., NC) implementations.
The geometry of graphs and some of its algorithmic applications
 COMBINATORICA
, 1995
"... In this paper we explore some implications of viewing graphs as geometric objects. This approach offers a new perspective on a number of graphtheoretic and algorithmic problems. There are several ways to model graphs geometrically and our main concern here is with geometric representations that res ..."
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Cited by 524 (19 self)
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In this paper we explore some implications of viewing graphs as geometric objects. This approach offers a new perspective on a number of graphtheoretic and algorithmic problems. There are several ways to model graphs geometrically and our main concern here is with geometric representations
Geometric variants of the Hofer norm
"... This note discusses some geometrically defined seminorms on the group Ham(M, ω) of Hamiltonian diffeomorphisms of a closed symplectic manifold (M, ω), giving conditions under which they are nondegenerate and explaining their relation to the Hofer norm. As a consequence we show that if an element in ..."
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Cited by 30 (1 self)
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This note discusses some geometrically defined seminorms on the group Ham(M, ω) of Hamiltonian diffeomorphisms of a closed symplectic manifold (M, ω), giving conditions under which they are nondegenerate and explaining their relation to the Hofer norm. As a consequence we show that if an element
Geometric Blur for Template Matching
, 2001
"... We address the problem of finding point correspondences in images by way of an approach to template matching that is robust under affine distortions. This is achieved by applying "geometric blur" to both the template and the image, resulting in a falloff in similarity that is close to lin ..."
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Cited by 149 (20 self)
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We address the problem of finding point correspondences in images by way of an approach to template matching that is robust under affine distortions. This is achieved by applying "geometric blur" to both the template and the image, resulting in a falloff in similarity that is close
Ranksparsity incoherence for matrix decomposition
, 2010
"... Suppose we are given a matrix that is formed by adding an unknown sparse matrix to an unknown lowrank matrix. Our goal is to decompose the given matrix into its sparse and lowrank components. Such a problem arises in a number of applications in model and system identification, and is intractable ..."
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Cited by 230 (21 self)
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to solve in general. In this paper we consider a convex optimization formulation to splitting the specified matrix into its components, by minimizing a linear combination of the ℓ1 norm and the nuclear norm of the components. We develop a notion of ranksparsity incoherence, expressed as an uncertainty
Sampling from large matrices: an approach through geometric functional analysis
 Journal of the ACM
, 2006
"... Abstract. We study random submatrices of a large matrix A. We show how to approximately compute A from its random submatrix of the smallest possible size O(r log r) with a small error in the spectral norm, where r = �A�2 F /�A�22 is the numerical rank of A. The numerical rank is always bounded by, a ..."
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Cited by 132 (5 self)
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Abstract. We study random submatrices of a large matrix A. We show how to approximately compute A from its random submatrix of the smallest possible size O(r log r) with a small error in the spectral norm, where r = �A�2 F /�A�22 is the numerical rank of A. The numerical rank is always bounded by
Approximation schemes for NPhard geometric optimization problems: A survey
 Mathematical Programming
, 2003
"... NPhard geometric optimization problems arise in many disciplines. Perhaps the most famous one is the traveling salesman problem (TSP): given n nodes in ℜ 2 (more generally, in ℜ d), find the minimum length path that visits each node exactly once. If distance is computed using the Euclidean norm (di ..."
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Cited by 44 (2 self)
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NPhard geometric optimization problems arise in many disciplines. Perhaps the most famous one is the traveling salesman problem (TSP): given n nodes in ℜ 2 (more generally, in ℜ d), find the minimum length path that visits each node exactly once. If distance is computed using the Euclidean norm
TENSOR RANK AND THE ILLPOSEDNESS OF THE BEST LOWRANK APPROXIMATION PROBLEM
"... There has been continued interest in seeking a theorem describing optimal lowrank approximations to tensors of order 3 or higher, that parallels the Eckart–Young theorem for matrices. In this paper, we argue that the naive approach to this problem is doomed to failure because, unlike matrices, te ..."
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Cited by 194 (13 self)
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, tensors of order 3 or higher can fail to have best rankr approximations. The phenomenon is much more widespread than one might suspect: examples of this failure can be constructed over a wide range of dimensions, orders and ranks, regardless of the choice of norm (or even Brègman divergence). Moreover
Approximating Geometric Optimization with l_pNorm Optimization
, 2000
"... In this article, we demonstrate how to approximate geometric optimization with l p  norm optimization. These two categories of problems are well known in structured convex optimization. We describe a family of l p norm optimization problems that can be made arbitrarily close to a geometric optimiz ..."
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In this article, we demonstrate how to approximate geometric optimization with l p  norm optimization. These two categories of problems are well known in structured convex optimization. We describe a family of l p norm optimization problems that can be made arbitrarily close to a geometric
A GEOMETRIC ESTIMATE ON THE NORM OF PRODUCT OF FUNCTIONALS
, 2006
"... Abstract. The open problem of determining the exact value of the nth linear polarization constant cn of R n has received considerable attention over the past few years. This paper makes a contribution to the subject by providing a new lower bound on the value of sup ‖y‖=1  〈x1,y 〉 · · · 〈xn,y 〉 ..."
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Cited by 5 (1 self)
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Abstract. The open problem of determining the exact value of the nth linear polarization constant cn of R n has received considerable attention over the past few years. This paper makes a contribution to the subject by providing a new lower bound on the value of sup ‖y‖=1  〈x1,y 〉 · · · 〈xn,y 〉 , where x1,...,xn are unit vectors in R n. The new estimate is given in terms of the eigenvalues of the Gram matrix [〈xi,xj〉] and improves upon earlier estimates of this kind. However, the intriguing conjecture cn = n n/2 remains open.
Results 1  10
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833