Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Constructive Discrepancy Minimization with Hereditary L2 Guarantees   In discrepancy minimization problems, we are given a family of sets
$\mathcal{S} = \{S_1,\dots,S_m\}$, with each $S_i \in \mathcal{S}$ a subset of
some universe $U = \{u_1,\dots,u_n\}$ of $n$ elements. The goal is to find a
coloring $\chi : U \to \{-1,+1\}$ of the elements of $U$ such that each set $S
\in \mathcal{S}$ is colored as evenly as possible. Two classic measures of
discrepancy are $\ell_\infty$-discrepancy defined as
$\textrm{disc}_\infty(\mathcal{S},\chi):=\max_{S \in \mathcal{S}} | \sum_{u_i
\in S} \chi(u_i) |$ and $\ell_2$-discrepancy defined as
$\textrm{disc}_2(\mathcal{S},\chi):=\sqrt{(1/|\mathcal{S}|)\sum_{S \in
\mathcal{S}} \left(\sum_{u_i \in S}\chi(u_i)\right)^2}$. Breakthrough work by
Bansal gave a polynomial time algorithm, based on rounding an SDP, for finding
a coloring $\chi$ such that $\textrm{disc}_\infty(\mathcal{S},\chi) = O(\lg n
\cdot \textrm{herdisc}_\infty(\mathcal{S}))$ where
$\textrm{herdisc}_\infty(\mathcal{S})$ is the hereditary
$\ell_\infty$-discrepancy of $\mathcal{S}$. We complement his work by giving a
simple $O((m+n)n^2)$ time algorithm for finding a coloring $\chi$ such
$\textrm{disc}_2(\mathcal{S},\chi) = O(\sqrt{\lg n} \cdot
\textrm{herdisc}_2(\mathcal{S}))$ where $\textrm{herdisc}_2(\mathcal{S})$ is
the hereditary $\ell_2$-discrepancy of $\mathcal{S}$. Interestingly, our
algorithm avoids solving an SDP and instead relies on computing
eigendecompositions of matrices. Moreover, we use our ideas to speed up the
Edge-Walk algorithm by Lovett and Meka [SICOMP'15]. To prove that our algorithm
has the claimed guarantees, we show new inequalities relating
$\textrm{herdisc}_\infty$ and $\textrm{herdisc}_2$ to the eigenvalues of the
matrix corresponding to $\mathcal{S}$. Our inequalities improve over previous
work by Chazelle and Lvov, and by Matousek et al. Finally, we also implement
our algorithm and show that it far outperforms random sampling.

26
The categories of this paper are: bayes-an
0
The categories of this paper are: hep-ph
21
The categories of this paper are: adap-org
24
The categories of this paper are: dg-ga
5
The categories of this paper are: gr-qc
9
The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th
23
The categories of this paper are: solv-int
28
The categories of this paper are: alg-geom
27
The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
11
The categories of this paper are: hep-lat
16
The categories of this paper are: stat
33
The categories of this paper are: plasm-ph
20
The categories of this paper are: acc-phys
29
The categories of this paper are: funct-an