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The Nature of Statistical Learning Theory
, 1999
"... Statistical learning theory was introduced in the late 1960’s. Until the 1990’s it was a purely theoretical analysis of the problem of function estimation from a given collection of data. In the middle of the 1990’s new types of learning algorithms (called support vector machines) based on the deve ..."
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Cited by 12976 (32 self)
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Statistical learning theory was introduced in the late 1960’s. Until the 1990’s it was a purely theoretical analysis of the problem of function estimation from a given collection of data. In the middle of the 1990’s new types of learning algorithms (called support vector machines) based
The hardness of the closest vector problem with preprocessing
 IEEE Transactions on Information Theory
, 2001
"... Abstract We give a new simple proof of the NPhardness of the closest vector problem. In addition to being much simpler than all previously known proofs, the new proof yields new interesting results about the complexity of the closest vector problem with preprocessing. This is a variant of the close ..."
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Cited by 43 (7 self)
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Abstract We give a new simple proof of the NPhardness of the closest vector problem. In addition to being much simpler than all previously known proofs, the new proof yields new interesting results about the complexity of the closest vector problem with preprocessing. This is a variant
Closest Vector Problem and the Closest Vector Problem with Preprocessing. The
, 2005
"... We present reductions from lattice problems in the ℓ2 norm to the corresponding problems in other norms such as ℓ1, ℓ ∞ (and in fact in any other ℓp norm where 1 ≤ p ≤ ∞). We consider ..."
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We present reductions from lattice problems in the ℓ2 norm to the corresponding problems in other norms such as ℓ1, ℓ ∞ (and in fact in any other ℓp norm where 1 ≤ p ≤ ∞). We consider
Covering cubes and the closest vector problem
 In Proceedings of the 27th annual ACM symposium on Computational geometry, SoCG ’11
, 2011
"... We provide the currently fastest randomized (1+ε)approximation algorithm for the closest lattice vector problem in the ℓ∞norm. The running time of our method depends on the dimension n and the approximation guarantee ε by 1 2 O(n) (log 1/ε) O(n) which improves upon the (2 + 1/ε) O(n) running time ..."
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Cited by 5 (0 self)
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We provide the currently fastest randomized (1+ε)approximation algorithm for the closest lattice vector problem in the ℓ∞norm. The running time of our method depends on the dimension n and the approximation guarantee ε by 1 2 O(n) (log 1/ε) O(n) which improves upon the (2 + 1/ε) O(n) running time
Lattice Sparsification and the Approximate Closest Vector Problem
, 2012
"... We give a deterministic algorithm for solving the (1 + ε) approximate Closest Vector Problem (CVP) on any n dimensional lattice and any norm in 2 O(n) (1 + 1/ε) n time and 2 n poly(n) space. Our algorithm builds on the lattice point enumeration techniques of Micciancio and Voulgaris (STOC 2010) and ..."
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Cited by 2 (2 self)
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We give a deterministic algorithm for solving the (1 + ε) approximate Closest Vector Problem (CVP) on any n dimensional lattice and any norm in 2 O(n) (1 + 1/ε) n time and 2 n poly(n) space. Our algorithm builds on the lattice point enumeration techniques of Micciancio and Voulgaris (STOC 2010
On the Closest Vector Problem with a Distance Guarantee
"... We present a substantially more efficient variant, both in terms of running time and size of preprocessing advice, of the algorithm by Liu, Lyubashevsky, and Micciancio [LLM06] for solving CVPP (the preprocessing version of the Closest Vector Problem, CVP) with a distance guarantee. For instance, fo ..."
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We present a substantially more efficient variant, both in terms of running time and size of preprocessing advice, of the algorithm by Liu, Lyubashevsky, and Micciancio [LLM06] for solving CVPP (the preprocessing version of the Closest Vector Problem, CVP) with a distance guarantee. For instance
Vector Problem and Closest Vector Problem in lattices.
"... Abstract. We provide unconditional constructions of concurrent statistical zeroknowledge proofs for a variety of nontrivial problems (not known to have probabilistic polynomialtime algorithms). The problems ..."
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Abstract. We provide unconditional constructions of concurrent statistical zeroknowledge proofs for a variety of nontrivial problems (not known to have probabilistic polynomialtime algorithms). The problems
Hardness of approximating the closest vector problem with preprocessing
 In FOCS
, 2005
"... Abstract We show that, unless NP`DTIME(2poly log(n)), the closest vector problem with preprocessing, for `p norm forany p> = 1, is hard to approximate within a factor of(log n)1/pffl for any ffl> 0. This improves the previous bestfactor of 3 1/p ffl due to Regev [19]. Our results also impl ..."
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Cited by 12 (1 self)
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Abstract We show that, unless NP`DTIME(2poly log(n)), the closest vector problem with preprocessing, for `p norm forany p> = 1, is hard to approximate within a factor of(log n)1/pffl for any ffl> 0. This improves the previous bestfactor of 3 1/p ffl due to Regev [19]. Our results also
Solving shortest and closest vector problems: The decomposition approach
"... Abstract. In this paper, we present a heuristic algorithm for solving exact, as well as approximate, SVP and CVP for lattices. This algorithm is based on a new approach which is very different from and complementary to the sieving technique. This new approach frees us from the kissing number bound a ..."
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Cited by 4 (1 self)
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and allows us to solve SVP and CVP in lattices of dimension n in time 2 0.377 n using memory 2 0.292 n. The key idea is to no longer work with a single lattice but to move the problems around in a tower of related lattices. We initiate the algorithm by sampling very short vectors in a dense overlattice
On Bounded Distance Decoding and the Closest Vector Problem with Preprocessing
"... We present a new efficient algorithm for the search version of the approximate Closest Vector Problem with Preprocessing (CVPP). This is the problem of finding a lattice vector whose distance from the target point is within some factor γ of the closest lattice vector, where the algorithm is allowed ..."
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We present a new efficient algorithm for the search version of the approximate Closest Vector Problem with Preprocessing (CVPP). This is the problem of finding a lattice vector whose distance from the target point is within some factor γ of the closest lattice vector, where the algorithm
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