k-SUM及相关问题的近优线性决策树
Near-optimal linear decision trees for k-SUM and related problems
摘要：
我们构建了近最优线性决策树的组合数学和离散几何的各种决策问题。例如，对于任何常数$k$，我们构造线性决策树，其使用$O(n \log^2 n)$线性查询来解决$n$元素上的$k$-SUM问题。此外，我们使用的查询是比较查询，它比较两个$k$ -子集的和;当被视为线性查询时，比较查询是$2k$ -稀疏的，并且只有$\{-1,0,1\}$系数。我们给出了类似的结构排序和集$A+B$和解决的SUBSET-SUM问题，无论是最佳数量的查询，多对数项。我们的建设是基于“推理维度”的概念，最近介绍了作者的背景下，积极的分类与比较查询。这可以被看作是对机器学习和离散几何之间富有成效的联系的另一个贡献，这可以追溯到VC维的发现。
英文摘要：
We construct near optimal linear decision trees for a variety of decision problems in combinatorics and discrete geometry. For example, for any constant $k$, we construct linear decision trees that solve the $k$-SUM problem on $n$ elements using $O(n \log^2 n)$ linear queries. Moreover, the queries we use are comparison queries, which compare the sums of two $k$-subsets; when viewed as linear queries, comparison queries are $2k$-sparse and have only $\{-1,0,1\}$ coefficients. We give similar constructions for sorting sumsets $A+B$ and for solving the SUBSET-SUM problem, both with optimal number of queries, up to poly-logarithmic terms. Our constructions are based on the notion of "inference dimension", recently introduced by the authors in the context of active classification with comparison queries. This can be viewed as another contribution to the fruitful link between machine learning and discrete geometry, which goes back to the discovery of the VC dimension.
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