Vector Balancing in Lebesgue Spaces

  title={Vector Balancing in Lebesgue Spaces},
  author={Victor Reis and Thomas Rothvoss},
A tantalizing conjecture in discrete mathematics is the one of Komlos, suggesting that for any vectors $\mathbf{a}_1,\ldots,\mathbf{a}_n \in B_2^m$ there exist signs $x_1, \dots, x_n \in \{ -1,1\}$ so that $\|\sum_{i=1}^n x_i\mathbf{a}_i\|_\infty \le O(1)$. It is a natural extension to ask what $\ell_q$-norm bound to expect for $\mathbf{a}_1,\ldots,\mathbf{a}_n \in B_p^m$. We prove that, for $2 \le p \le q \le \infty$, such vectors admit fractional colorings $x_1, \dots, x_n \in [-1,1]$ with a… 

A new framework for matrix discrepancy: partial coloring bounds via mirror descent

The matrix Spencer conjecture is reduced to the existence of a O(log(m/n) quantum relative entropy net on the spectraplex using a Gaussian measure lower bound of 2−O(n) for a scaling of the discrepancy body.

Improved Algorithms for Combinatorial Discrepancy

Discrepancy theory is a subfield of combinatorics which has branched in Computer Science due to several connections it has to geometric problems, randomized algorithms and complexity theory [13, 8].

Hyperbolic Concentration, Anti-Concentration, and Discrepancy

This paper presents the first work that shows either concentration or anti-concentration results for hyperbolic polynomials and implies that a random bi-coloring of any set system with n sets and n elements will have discrepancy O ( √ n log n ) with high probability.

Resolving Matrix Spencer Conjecture Up to Poly-logarithmic Rank

A simple proof of the matrix Spencer conjecture up to poly-logarithmic rank is given, which implies a log n − Ω(log log n ) qubit lower bound for quantum random access codes encoding n classical bits with advantage ≫ 1 / √ n .



Constructive Algorithms for Discrepancy Minimization

  • N. Bansal
  • Mathematics, Computer Science
    2010 IEEE 51st Annual Symposium on Foundations of Computer Science
  • 2010
The main idea in the algorithms is to produce a coloring over time by letting the color of the elements perform a random walk (with tiny increments) starting from 0 until they reach $\pm 1$.

Constructive Discrepancy Minimization for Convex Sets

  • T. Rothvoss
  • Mathematics
    2014 IEEE 55th Annual Symposium on Foundations of Computer Science
  • 2014
It is shown that for any symmetric convex set K with measure at least e-n/500, the following algorithm finds a point y ∈ K ∩ [-1, 1]n with Ω(n) coordinates in ±1: (1) take a random Gaussian vector x, (2) compute the point y in K∩ [- 1, 1)n that is closest to x.

Constructive Discrepancy Minimization by Walking on the Edges

A new randomized algorithm to find a coloring as in Spencer's result based on a restricted random walk which is “truly” constructive in that it does not appeal to the existential arguments, giving a new proof of Spencer's theorem and the partial coloring lemma.

A Matrix Hyperbolic Cosine Algorithm and Applications

This paper generalizes Spencer's hyperbolic cosine algorithm to the matrix-valued setting, and gives an elementary connection between spectral sparsification of positive semi-definite matrices and element-wise matrixSparsification, which implies an improved deterministic algorithm for spectral graph sparsify of dense graphs.

Efficient algorithms for discrepancy minimization in convex sets

A constructive version of the result of Gluskin and Giannopoulos, in which the coloring is attained via the optimization of a linear function is proved, which implies a linear programming based algorithm for combinatorial discrepancy obtaining the same result as Spencer.

Six standard deviations suffice

Given n sets on n elements it is shown that there exists a two-coloring such that all sets have discrepancy at most Knl/2, K an absolute constant. This improves the basic probabilistic method with

Balancing Vectors in Any Norm

All vector balancing constants admit "good" approximate characterizations, with approximation factors depending only polylogarithmically on the dimension n, and a novel convex program is presented which encodes the "best possible way" to apply Banaszczyk's vector balancing theorem for boundingvector balancing constants from above.

The Gram-Schmidt walk: a cure for the Banaszczyk blues

This paper gives an efficient randomized algorithm to find a ± 1 combination of the vectors which lies in cK for c>0 an absolute constant, which leads to new efficient algorithms for several problems in discrepancy theory.

Geometric Algorithms and Combinatorial Optimization

0. Mathematical Preliminaries.- 0.1 Linear Algebra and Linear Programming.- Basic Notation.- Hulls, Independence, Dimension.- Eigenvalues, Positive Definite Matrices.- Vector Norms, Balls.- Matrix

Convex bodies with few faces

It is proved that if u, u,n are vectors in R, k r. An application to number theory is stated. 0. INTRODUCTION In [V], Vaaler proved that if Q, = [v I]n is the central unit cube in Rn and U is a