Filters








17 Hits in 1.2 sec

Towards a Constructive Version of Banaszczyk's Vector Balancing Theorem [article]

Daniel Dadush, Shashwat Garg, Shachar Lovett, Aleksandar Nikolov
2016 arXiv   pre-print
A major open problem is to devise a constructive version of Banaszczyk's vector balancing theorem, i.e. to find an efficient algorithm which constructs the signed combination.  ...  a signed combination of these vectors which lands inside K.  ...  Acknowledgments We would like to thank the American Institute for Mathematics for hosting a recent workshop on discrepancy theory, where some of this work was done.  ... 
arXiv:1612.04304v1 fatcat:7a6rhwnzpbf7bfri33p74whl3m

Towards a Constructive Version of Banaszczyk's Vector Balancing Theorem

Daniel Dadush, Shashwat Garg, Shachar Lovett, Aleksandar Nikolov
unpublished
A major open problem is to devise a constructive version of Banaszczyk's vector balancing theorem, i.e. to find an efficient algorithm which constructs the signed combination.  ...  a signed combination of these vectors which lands inside K.  ...  We would like to thank the American Institute for Mathematics for hosting a recent workshop on discrepancy theory, where some of this work was done.  ... 
fatcat:zawcxxygafhzdmznenbeugrvne

The gram-schmidt walk: a cure for the Banaszczyk blues

Nikhil Bansal, Daniel Dadush, Shashwat Garg, Shachar Lovett
2018 Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2018  
Banaszczyk's proof of this result is non-constructive and a major open problem has been to give an efficient algorithm to find such a ±1 combination of the vectors.  ...  An important result in discrepancy due to Banaszczyk states that for any set of n vectors in R m of ℓ 2 norm at most 1 and any convex body K in R m of Gaussian measure at least half, there exists a ±1  ...  In a recent result, [DGLN16] reformulated Banaszczyk's result in terms of certain subgaussian distributions and reduced the question of finding an algorithmic version of Banaszczyk's result to constructing  ... 
doi:10.1145/3188745.3188850 dblp:conf/stoc/BansalDGL18 fatcat:l554rhdiefgmjler5cri2blhlq

The Gram-Schmidt Walk: A Cure for the Banaszczyk Blues [article]

Nikhil Bansal, Daniel Dadush, Shashwat Garg, Shachar Lovett
2017 arXiv   pre-print
Banaszczyk's proof of this result is non-constructive and a major open problem has been to give an efficient algorithm to find such a ± 1 combination of the vectors.  ...  An important result in discrepancy due to Banaszczyk states that for any set of n vectors in R^m of ℓ_2 norm at most 1 and any convex body K in R^m of Gaussian measure at least half, there exists a ± 1  ...  In a recent result, [DGLN16] reformulated Banaszczyk's result in terms of certain subgaussian distributions and reduced the question of finding an algorithmic version of Banaszczyk's result to constructing  ... 
arXiv:1708.01079v1 fatcat:mxedj45aifek7bsaky4dni2soy

Gaussian discrepancy: a probabilistic relaxation of vector balancing [article]

Sinho Chewi, Patrik Gerber, Philippe Rigollet, Paxton Turner
2022 arXiv   pre-print
We show that Gaussian discrepancy is a tighter relaxation than the previously studied vector and spherical discrepancy problems, and we construct a fast online algorithm that achieves a version of the  ...  We introduce a novel relaxation of combinatorial discrepancy called Gaussian discrepancy, whereby binary signings are replaced with correlated standard Gaussian random variables.  ...  SC was supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program.  ... 
arXiv:2109.08280v2 fatcat:dcxtmd37k5e2xglfwszdomdyka

Approximating Hereditary Discrepancy via Small Width Ellipsoids [chapter]

Aleksandar Nikolov, Kunal Talwar
2014 Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms  
The Hereditary Discrepancy of a hypergraph, defined as the maximum discrepancy of a restriction of the hypergraph to a subset of its vertices, is a measure of its complexity.  ...  of any point in a ellipsoid containing the columns of A.  ...  Acknowledgements The first named author thanks Daniel Dadush for useful discussions, and Assaf Naor for bringing up the question about a geometric equivalent of Lemma 16.  ... 
doi:10.1137/1.9781611973730.24 dblp:conf/soda/NikolovT15 fatcat:d6yyaykggffrnfvwy647yldmai

Approximating Hereditary Discrepancy via Small Width Ellipsoids [article]

Aleksandar Nikolov, Kunal Talwar
2014 arXiv   pre-print
The Hereditary Discrepancy of a hypergraph, defined as the maximum discrepancy of a restriction of the hypergraph to a subset of its vertices, is a measure of its complexity.  ...  of any point in a ellipsoid containing the columns of A.  ...  Acknowledgements The first named author thanks Daniel Dadush for useful discussions, and Assaf Naor for bringing up the question about a geometric equivalent of Lemma 16.  ... 
arXiv:1311.6204v2 fatcat:sgoa4xsgyrdc5pvyvnj2jts57a

Discrepancy Minimization via a Self-Balancing Walk [article]

Ryan Alweiss, Yang P. Liu, Mehtaab Sawhney
2020 arXiv   pre-print
We study discrepancy minimization for vectors in R^n under various settings. The main result is the analysis of a new simple random process in multiple dimensions through a comparison argument.  ...  As corollaries, we obtain bounds which are tight up to logarithmic factors for several problems in online vector balancing posed by Bansal, Jiang, Singla, and Sinha (STOC 2020), as well as linear time  ...  We thank Nihkil Bansal for observations which led to improvements of several logarithmic factors in Theorem 3.1 and Theorem 3.2. R.A. is supported by an NSF Graduate Research Fellowship.  ... 
arXiv:2006.14009v2 fatcat:3hn664zfvvgbveokmplr3ybsnu

A Unified Approach to Discrepancy Minimization [article]

Nikhil Bansal, Aditi Laddha, Santosh S. Vempala
2022 arXiv   pre-print
We study a unified approach and algorithm for constructive discrepancy minimization based on a stochastic process.  ...  We demonstrate the flexibility of the method by deriving a discrepancy bound for smoothed instances, which interpolates between known bounds for worst-case and random instances.  ...  In further progress, an algorithmic version of the O( √ log n) bound for the Komlós problem was obtained by [4] , see also [6] , and [5] for the more general algorithmic version of Banaszczyk's result  ... 
arXiv:2205.01023v1 fatcat:ptrgl7toyzcr3fn7lop3ld5txq

Random Self-reducibility of Ideal-SVP via Arakelov Random Walks [chapter]

Koen de Boer, Léo Ducas, Alice Pellet-Mary, Benjamin Wesolowski
2020 Lecture Notes in Computer Science  
We then exploit this result to relate the average-case and the worst-case of the Shortest Vector Problem in ideal lattices.  ...  Fixing a number field, the space of all ideal lattices, up to isometry, is naturally an abelian group, called the Arakelov class group.  ...  The proof of this theorem is available in the full version [7] . Lemma 2 . 2 ( 22 Sampling of prime ideals, ERH).  ... 
doi:10.1007/978-3-030-56880-1_9 fatcat:zbnpezz2kfdf3dlnhklqwgfwea

Discrepancy Without Partial Colorings

Nicholas J. A. Harvey, Roy Schwartz, Mohit Singh, Marc Herbstritt
2014 International Workshop on Approximation Algorithms for Combinatorial Optimization  
Spencer's non-constructive theorem is used to ensure feasibility of each SDP.  ...  For Bansal's algorithm, this is because feasibility of the SDP is proven using Spencer's theorem, which only ensures existence of a "partial vector coloring", not a "full vector coloring".  ...  Appendix A Proof of Corollary 1 Proof. The proof follows the same outline but differs slightly from the proof of Theorem 2.  ... 
doi:10.4230/lipics.approx-random.2014.258 dblp:conf/approx/HarveySS14 fatcat:ztvucy7jcndshaqgue3aremogq

TIGHTER BOUNDS FOR THE DISCREPANCY OF BOXES AND POLYTOPES

Aleksandar Nikolov
2017 Mathematika  
Combinatorial discrepancy is a complexity measure of a collection of sets which quantifies how well the sets in the collection can be simultaneously balanced.  ...  by dilations and translations of a fixed convex polytope $B$ .  ...  The author would like to thank the organizers of the 2016 discrepancy theory workshop in Varenna, dedicated to the memory of Klaus Roth and Jiří Matoušek, where some of the initial ideas in this paper  ... 
doi:10.1112/s0025579317000250 fatcat:rch5qjmy3rej7cfcbapbuhriei

Tighter Bounds for the Discrepancy of Boxes and Polytopes [article]

Aleksandar Nikolov
2017 arXiv   pre-print
Combinatorial discrepancy is a complexity measure of a collection of sets which quantifies how well the sets in the collection can be simultaneously balanced.  ...  We improve the best known upper bound for the Tusnady problem by a logarithmic factor, using a result of Banaszczyk on signed series of vectors.  ...  The author would like to thank the organizers of the 2016 discrepancy theory workshop in Varenna, where some of the initial ideas in this paper were conceived.  ... 
arXiv:1701.05532v2 fatcat:gbxcukuxvzdk3gswqoony3dqe4

Factorization Norms and Hereditary Discrepancy [article]

Jiri Matousek, Aleksandar Nikolov, Kunal Talwar
2015 arXiv   pre-print
The γ_2 norm of a real m× n matrix A is the minimum number t such that the column vectors of A are contained in a 0-centered ellipsoid E⊆R^m which in turn is contained in the hypercube [-t, t]^m.  ...  Most notably, we prove a new lower bound of Ω(^d-1 n) for the d-dimensional Tusnády problem, asking for the combinatorial discrepancy of an n-point set in R^d with respect to axis-parallel boxes.  ...  Acknowledgments We would like to thank Alan Edelman and Gil Strang for invaluable advice concerning the singular values of the matrix in Proposition 4.1, and Van Vu for recommending the right experts for  ... 
arXiv:1408.1376v2 fatcat:u6yktvidtjcpxkic7c6uvkpp7i

A Gaussian Fixed Point Random Walk

Yang P. Liu, Ashwin Sah, Mehtaab Sawhney, Mark Braverman
2022
As an immediate corollary, we obtain an online version of Banaszczyk's discrepancy result for partial colorings and ±1,2 signings.  ...  In this note, we design a discrete random walk on the real line which takes steps 0,±1 (and one with steps in {±1,2}) where at least 96% of the signs are ±1 in expectation, and which has 𝒩(0,1) as a stationary  ...  /2] (1 − r 1 (f )) ≥ 0.9639. ◀ Our second algorithmic application recovers the online vector balancing results of Alweiss, Liu, and Sawhney [1, Theorems 1.1, 1.2]. 101:7 ▶ Theorem 12.  ... 
doi:10.4230/lipics.itcs.2022.101 fatcat:vgwdws4sk5hjznc55zx5ybnskq
« Previous Showing results 1 — 15 out of 17 results