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Improved Algorithms for the Approximate k-List Problem in Euclidean Norm [chapter]

Gottfried Herold, Elena Kirshanova
2017 Lecture Notes in Computer Science  
We present an algorithm for the approximate k-List problem for the Euclidean distance that improves upon the Bai-Laarhoven-Stehlé (BLS) algorithm from ANTS'16.  ...  The improvement stems from the observation that almost all the solutions to the approximate k-List problem form a particular configuration in n-dimensional space.  ...  We would like to thank the authors of [4] , Shi Bai, Damien Stehlé, and Thijs Laarhoven for constructive discussions.  ... 
doi:10.1007/978-3-662-54365-8_2 fatcat:iqdtbhlkgngiphmqogofiez3oq

Norm-Explicit Quantization: Improving Vector Quantization for Maximum Inner Product Search [article]

Xinyan Dai, Xiao Yan, Kelvin K. W. Ng, Jie Liu, James Cheng
2019 arXiv   pre-print
NEQ quantizes the norms of items in a dataset explicitly to reduce errors in norm, which is crucial for MIPS.  ...  The experimental results show that NEQ improves the performance of various VQ techniques for MIPS, including PQ, OPQ, RQ and AQ.  ...  NEQ solves this problem with the codebook explicitly reduce the error in norm because accurate norm is learning process in Algorithm 2. important for MIPS.  ... 
arXiv:1911.04654v2 fatcat:kna73kr5enbgvi6grflolitypm

Faster provable sieving algorithms for the Shortest Vector Problem and the Closest Vector Problem on lattices in ℓ_p norm [article]

Priyanka Mukhopadhyay
2019 arXiv   pre-print
In this paper we give provable sieving algorithms for the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP) on lattices in ℓ_p norm for 1≤ p≤∞.  ...  This improves the running time, specially in the ℓ_2 norm, where we achieve a time complexity of 2^2.25n+o(n), while the List Sieve Birthday algorithm [Pujol an Stehle, 2009] has a running time 2^2.465n  ...  The author would also like to thank Artur Mariano for clarifying some results in [MLB17] .  ... 
arXiv:1907.04406v2 fatcat:wre6mpspmbacnemtok7lehy6qi

Page 7893 of Mathematical Reviews Vol. , Issue 99k [page]

1999 Mathematical Reviews  
“We also give similar approximation schemes for some other NP-hard Euclidean problems: minimum Steiner tree, k-TSP, and k-MST.  ...  The previous best approximation algorithm for the problem (due to Christofides) achieves a 3/2-approximation in polynomial time.  ... 

Faster Provable Sieving Algorithms for the Shortest Vector Problem and the Closest Vector Problem on Lattices in ℓp Norm

Priyanka Mukhopadhyay
2021 Algorithms  
In this work, we give provable sieving algorithms for the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP) on lattices in ℓp norm (1≤p≤∞).  ...  This improves the running time, especially in the ℓ2 norm, where we achieve a time complexity of 22.25n+o(n), while the List Sieve Birthday algorithm has a running time of 22.465n+o(n).  ...  Acknowledgments: The author would like to acknowledge the anonymous reviewers for their helpful comments that have helped to improve the manuscript significantly.  ... 
doi:10.3390/a14120362 fatcat:2brqjjy76jabjkg4th4rfsebse

Approximate CVP in time 2^0.802 n – now in any norm! [article]

Thomas Rothvoss, Moritz Venzin
2021 arXiv   pre-print
For both problems, SVP and CVP, we reduce to the case of the Euclidean norm.  ...  We show that a constant factor approximation of the shortest and closest lattice vector problem in any norm can be computed in time 2^0.802 n.  ...  time algorithm for a constant factor approximation to the closest vector problem in any norm.  ... 
arXiv:2110.02387v1 fatcat:wbj262rakreqxdal57ibpavak4

Improved Algorithms for the Shortest Vector Problem and the Closest Vector Problem in the Infinity Norm

Divesh Aggarwal, Priyanka Mukhopadhyay, Michael Wagner
2018 International Symposium on Algorithms and Computation  
As in [6, 11] , we also extend this algorithm to obtain significantly faster algorithms for approximate versions of the shortest vector problem and the closest vector problem (CVP) in thenorm.  ...  We give a new sieving procedure that runs in time linear in N , thereby improving the running time of the algorithm for SVP in thenorm.  ...  Acknowledgements We thank the anonymous referees who helped improve the draft of this paper.  ... 
doi:10.4230/lipics.isaac.2018.35 dblp:conf/isaac/AggarwalM18 fatcat:23h37gqehjemvhuufnnqqxjfi4

Fast bregman divergence NMF using taylor expansion and coordinate descent

Liangda Li, Guy Lebanon, Haesun Park
2012 Proceedings of the 18th ACM SIGKDD international conference on Knowledge discovery and data mining - KDD '12  
Most of the algorithms proposed in literature for NMF have been based on minimizing the Frobenius norm.  ...  This is partly due to the fact that the minimization problem based on the Frobenius norm provides much more flexibility in algebraic manipulation than other divergences.  ...  [5] introduced an alternative algorithm with improved local updating rules, achieving high efficiency in solving NMF problems. The resulting algorithm applies to the Frobenius norm.  ... 
doi:10.1145/2339530.2339582 dblp:conf/kdd/LiLP12 fatcat:k2s55blelfeihkxdrbwc6puegy

Nearest Neighbor Search for Hyperbolic Embeddings [article]

Xian Wu, Moses Charikar
2020 arXiv   pre-print
We focus on the problem of nearest neighbor search, a fundamental problem in data analysis.  ...  We present efficient algorithmic solutions that build upon established methods for nearest neighbor search in Euclidean space, allowing for easy adoption and integration with existing systems.  ...  Our second main class of algorithms uses the insight that when p ∈ D have similar Euclidean norms, the denominator term (1− p 2 )(1− q 2 ) in Eq. 1 is similar for different p, so the problem reduces to  ... 
arXiv:2009.00836v1 fatcat:k6ozjfmjtzdxlirispyoxn3ukq

Approximate CVP_p in Time 2^{0.802 n}

Friedrich Eisenbrand, Moritz Venzin, Peter Sanders, Grzegorz Herman, Fabrizio Grandoni
2020 European Symposium on Algorithms  
This matches the currently fastest constant factor approximation algorithm for the shortest vector problem w.r.t. 𝓁₂.  ...  We show that a constant factor approximation of the shortest and closest lattice vector problem w.r.t. any 𝓁_p-norm can be computed in time 2^{(0.802 +ε) n}.  ...  This directly improves the running time of the algorithms for p norms that hinge on the kissing number.  ... 
doi:10.4230/lipics.esa.2020.43 dblp:conf/esa/EisenbrandV20 fatcat:jj6xo272fjcebcqojty6omkcpa

Approximate CVP_p in time 2^0.802 n [article]

Friedrich Eisenbrand, Moritz Venzin
2020 arXiv   pre-print
This matches the currently fastest constant factor approximation algorithm for the shortest vector problem w.r.t. ℓ_2.  ...  We show that a constant factor approximation of the shortest and closest lattice vector problem w.r.t. any ℓ_p-norm can be computed in time 2^(0.802 +ϵ) n.  ...  This directly improves the running time of the algorithms for p norms that hinge on the kissing number.  ... 
arXiv:2005.04957v2 fatcat:ws4v7nc35baincg5ebgbfix6q4

Computation of the Euclidean minimum of algebraic number fields

Pierre Lezowski
2013 Mathematics of Computation  
Then, we show how to apply the algorithm to study extensions of norm-Euclideanity. We consider an algebraic number field K. Let Z K be its ring of integers.  ...  of norm-Euclidean or non-norm-Euclidean algebraic number fields.  ...  Acknowledgements I am very grateful to Jean-Paul Cerri for his invaluable help at each step of the redaction of this paper.  ... 
doi:10.1090/s0025-5718-2013-02746-9 fatcat:hdgxzmjlbvbnjorcwfpp6apacq

Tuning GaussSieve for Speed [chapter]

Robert Fitzpatrick, Christian Bischof, Johannes Buchmann, Özgür Dagdelen, Florian Göpfert, Artur Mariano, Bo-Yin Yang
2015 Lecture Notes in Computer Science  
In this paper, we propose a number of heuristic improvements to GaussSieve, which can also be applied to other sieving algorithms for SVP.  ...  With contemporary lattice-based cryptographic proposals relying largely on the hardness of solving the shortest and closest vector problems in ideal lattices, examining possible improvements to sieving  ...  Acknowledgments The authors would like to thank the anonymous reviewers of Latincrypt 2014 for their helpful comments and suggestions which substantially improved this paper.  ... 
doi:10.1007/978-3-319-16295-9_16 fatcat:wv3eqv7t3rfpnnp5d2y33ehcqy

Noisy Chinese remaindering in the Lee norm

Igor E. Shparlinski, Ron Steinfeld
2004 Journal of Complexity  
small additive noise bounded in the Lee norm.  ...  We use lattice reduction to obtain a polynomial time algorithm for recovering an integer (up to a small interval) from its residues modulo sufficiently many primes, when the residues are corrupted by a  ...  We remark that our algorithm for the Lee norm uses lattice basis reduction techniques more directly than the list decoding algorithm for the Hamming norm [8] , which first transforms the problem to an  ... 
doi:10.1016/j.jco.2003.08.020 fatcat:ni2y5ty2ijejhlxq2k5j4tgzfq

A Deterministic Single Exponential Time Algorithm for Most Lattice Problems Based on Voronoi Cell Computations

Daniele Micciancio, Panagiotis Voulgaris
2013 SIAM journal on computing (Print)  
This improves the n O(n) running time of the best previously known algorithms for CVP (Kannan, Math.  ...  In the process, we also give algorithms for several other lattice problems, including computing the kissing number of a lattice, and computing the set of all Voronoi relevant vectors.  ...  No NP-hardness proof for the covering radius problem in the 2 norm is known (but see [26] for NPhardness results in p norm for large p).  ... 
doi:10.1137/100811970 fatcat:t7siwuxhv5fhzf7o2w773jcfba
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