Constructive Discrepancy Minimization with Hereditary L2 Guarantees

被引:3
|
作者
Larsen, Kasper Green [1 ]
机构
[1] Aarhus Univ, Dept Comp Sci, Aarhus, Denmark
关键词
Discrepancy; Hereditary Discrepancy; Combinatorics; Computational Geometry; BOUNDS;
D O I
10.4230/LIPIcs.STACS.2019.48
中图分类号
TP301 [理论、方法];
学科分类号
081202 ;
摘要
In discrepancy minimization problems, we are given a family of sets S =,S,A, with each Si E S a subset of some universe U =, un} of n elements. The goal is to find a coloring x : U {-1, +1} of the elements of U such that each set S E S is colored as evenly as possible. Two classic measures of discrepancy are Coo -discrepancy defined as disco, (S, x) := maxsEs E,,Es x(ui)I 2 and p2 -discrepancy defined as disc2 (S, x) := (1/ IS I,E,ses Vui)). Breakthrough work by Bansal [FOCS'10] gave a polynomial time algorithm, based on rounding an SDP, for finding a coloring x such that disc (S, x) = 0(1g n " herdisc,,(S)) where herdisc,,(S) is the hereditary Coo -discrepancy of S. We complement his work by giving a clean and simple 0((m n)n2) time algorithm for finding a coloring x such disc2(S, x) = 0( \/1g n " herdisc2(S)) where herdisc2(S) is the hereditary ?2 -discrepancy of S. Interestingly, our algorithm avoids solving an SDP and instead relies simply on computing eigendecompositions of matrices. To prove that our algorithm has the claimed guarantees, we also prove new inequalities relating both herdisc,, and herdisc2 to the eigenvalues of the incidence matrix corresponding to S. Our inequalities improve over previous work by Chazelle and Lvov [SCG'00] and by Matousek, Nikolov and Talwar [SODA'15+SCG'15]. We believe these inequalities are of independent interest as powerful tools for proving hereditary discrepancy lower bounds. Finally, we also implement our algorithm and show that it far outperforms random sampling of colorings in practice. Moreover, the algorithm finishes in a reasonable amount of time on matrices of sizes up to 10000 x 10000.
引用
收藏
页数:13
相关论文
共 50 条
  • [1] Constructive Algorithms for Discrepancy Minimization
    Bansal, Nikhil
    [J]. 2010 IEEE 51ST ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE, 2010, : 3 - 10
  • [2] CONSTRUCTIVE DISCREPANCY MINIMIZATION BY WALKING ON THE EDGES
    Lovett, Shachar
    Meka, Raghu
    [J]. SIAM JOURNAL ON COMPUTING, 2015, 44 (05) : 1573 - 1582
  • [3] Constructive discrepancy minimization for convex sets
    Rothvoss, Thomas
    [J]. 2014 55TH ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS 2014), 2014, : 140 - 145
  • [4] Constructive Discrepancy Minimization by Walking on The Edges
    Lovett, Shachar
    Meka, Raghu
    [J]. 2012 IEEE 53RD ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS), 2012, : 61 - 67
  • [5] CONSTRUCTIVE DISCREPANCY MINIMIZATION FOR CONVEX SETS
    Rothvoss, Thomas
    [J]. SIAM JOURNAL ON COMPUTING, 2017, 46 (01) : 224 - 234
  • [6] A Sublinear Algorithm for Sparse Reconstruction with l2/l2 Recovery Guarantees
    Calderbank, Robert
    Howard, Stephen
    Jafarpour, Sina
    [J]. 2009 3RD IEEE INTERNATIONAL WORKSHOP ON COMPUTATIONAL ADVANCES IN MULTI-SENSOR ADAPTIVE PROCESSING (CAMSAP 2009), 2009, : 209 - +
  • [7] STRUCTURAL AND CONSTRUCTIVE CHARACTERISTICS OF FUNCTIONS IN L2
    TAIKOV, LV
    [J]. MATHEMATICAL NOTES, 1979, 25 (1-2) : 113 - 116
  • [8] Visual Tracking Using L2 Minimization
    Pei, Zhijun
    Han, Lei
    [J]. INTERNATIONAL SEMINAR ON APPLIED PHYSICS, OPTOELECTRONICS AND PHOTONICS (APOP 2016), 2016, 61
  • [9] L2 discrepancy of generalized Zaremba point sets
    Faure, Henri
    Pillichshammer, Friedrich
    [J]. JOURNAL DE THEORIE DES NOMBRES DE BORDEAUX, 2011, 23 (01): : 121 - 136
  • [10] Accelerated schemes for the L1/L2 minimization
    Wang, Chao
    Yan, Ming
    Rahimi, Yaghoub
    Lou, Yifei
    [J]. IEEE Transactions on Signal Processing, 2020, 68 : 2660 - 2669