We give an efficient algorithm with improved algorithmic guarantees for the (offline) Beck--Fiala problem when the sets have bounded size. Let $A$ be an arbitrary matrix $A\in\{0,1\}^{m\times n}$ with at most $d$ ones per column and at most $s$ ones per row. Let $\log^*$ denote the iterated logarithm and $\ell_j$ denote the $j$-fold composition of log. Assume $s\le\exp(O(\sqrt d))$. We provide an efficient algorithm that, for arbitrary sparsity $d$, gives $O(\sqrt d(1+\log^*n))$ discrepancy. Moreover, if $d\ge\ell_j(n)$ for a fixed integer $j\ge1$, the algorithm gives $O_j(\sqrt d)$ discrepancy. The proof is a bootstrapping scheme using the Bansal-Jiang algorithm.
Let $A, B \in \mathbb{Z}_{\ge 0}^n$ be nonnegative vectors and let $t = |\operatorname{supp}(A \star B)|$. We give a Las Vegas algorithm that computes $A \star B$ in $O(t \log t)$ expected time. More generally, for every $0<\delta \le \frac{1}{2}$, the algorithm terminates within $O(t \log t \log \frac{1}{\delta})$ tim...
A deterministic algorithm that finds signs with $\|A\varepsilon\|_\infty<99$ using $O(mn+n^{\omega+2}\log^3 n)$ arithmetic operations, where $\omega>2$ is any fixed attainable matrix-multiplication exponent; with the current bounds on $\omega$ this is $\widetilde O(mn+n^{4.372})$.
Let $\pi(\mathrm{d} x)\propto e^{-U(x)}\, \mathrm{d} x$ on $\mathbb{R}^d$, where $U$ is continuously differentiable and $m$-strongly convex with a globally $L$-Lipschitz gradient, $0<m\leq L<\infty$, and $\kappa=L/m$. Fixed-step Metropolis-adjusted Langevin algorithm (MALA) has known warm-start mixing-time upper bounds...
Harwit and Sloane conjectured that every nonsingular entrywise-nonnegative matrix $A\in\mathbb R^{n\times n}$ satisfies $\|A^{-1}\|_F\ge 2n(n+1)^{-1}\|A\|_{\max}^{-1}$, with equality precisely for positive multiples of $S$-matrices. Cheng proved the conjecture in odd dimensions, while Frankel and Urschel proved the eve...
Let $P_n$ be the matrix of a random permutation of $n$ symbols and let $M_n=\log\max_{|z|=1}|\det(I-zP_n)|$. Cook and Zeitouni proved that $M_n/\log n$ converges in probability to a constant $x_0$ for a uniform permutation. We show that the $\sqrt{\log n}$ fluctuations of $M_n$ are carried entirely by the number of cyc...
We give a random-bit-efficient construction for the inverse star discrepancy. For every fixed $u\in(0,1)$, $k$-wise independent uniform points $\boldsymbol{X}_1,\ldots,\boldsymbol{X}_N$ with $k=O(d(1+\log(1+N/d)))$ satisfy the Monte Carlo bound $D_N^*(\boldsymbol{X}_1,\ldots,\boldsymbol{X}_N) =O(\sqrt{d/N})$ with proba...
Kosuke Suzuki· 0 citations
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