AI 中文总结
该研究分析有限点集对超平面等仿射代数集的红-蓝偏差，证明偏差界较VC维界有多项式改进，构造了偏差下界并将方法应用于通信复杂性的代价分离问题。
AI 中文摘要
我们研究有限点集相对于超平面以及更一般的有界复杂度仿射代数集的组合（红-蓝）偏差。我们证明，实欧氏空间中的每个n点集都存在一种红-蓝着色，使得每个维数至多为D、次数至多为k的仿射代数集的偏差至多为n^(1/2 - 1/(2(D+1)) - ε)，其中ε=ε(D,k)>0。这比直接的VC维界Õ(n^(1/2 - 1/(2(D+1))))有多项式级改进。反之，我们在ℝ^d中构造了n点集，其相对于超平面的偏差为Ω̃(n^(1/2 - 1/(d+1)))，推广了Chazelle和Lvov的点-线偏差下界。我们还介绍了该方法在通信复杂性中的进一步应用，涉及访问相等预言机时随机通信代价与确定性通信代价之间的分离。
英文摘要
We study the combinatorial (red-blue) discrepancy of finite point sets with respect to hyperplanes and, more generally, bounded-complexity affine algebraic sets. We prove that every $n$-point set in a real Euclidean space admits a red-blue coloring for which every affine algebraic set of dimension at most $D$ and degree at most $k$ has discrepancy at most $n^{\frac12-\frac{1}{2(D+1)}-\varepsilon}$ for some $\varepsilon=\varepsilon(D,k)>0$. This gives a polynomial improvement over the straightforward VC-dimension bound $\tilde O(n^{\frac12-\frac{1}{2(D+1)}})$. In the opposite direction, we construct $n$-point sets in $\mathbb R^d$ whose discrepancy with respect to hyperplanes is $\tildeΩ(n^{\frac12-\frac{1}{d+1}}),$ extending the point-line discrepancy lower bound of Chazelle and Lvov. We present further applications of our methods in communication complexity, concerning separation between randomized communication cost and deterministic communication cost with access to equality oracle.
Comments32 pages, 1 figure
