发表机构
NYU; University of Pennsylvania（纽约大学; 宾夕法尼亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
从数据驱动算法设计角度研究最近邻搜索，专注平衡半空间树。在类似高斯边际条件下给出算法，能学习到\(o(nd)\)查询时间的树。核心是平衡半空间切割问题，设计非恰当学习算法规避NP难，输出特定平衡多项式阈值函数。
AI 中文摘要
我们从数据驱动算法设计的角度研究最近邻搜索：给定一个大小为\(n\)的数据集\(P\subset\mathbb{R}^d\)以及对\(\mathbb{R}^d\)上查询分布的样本访问，目标是学习一种针对从该特定分布抽取的查询进行优化的数据结构。我们专注于平衡半空间树类，它自然地抽象了诸如局部敏感哈希之类的空间划分框架。假设数据集和查询分布具有类似高斯的边际条件，我们给出一种高效算法，若存在完美树，则该算法能学习到一棵实现\(o(nd)\)查询时间的树。我们算法方法的核心是平衡半空间切割问题，即在给定\(\mathbb{R}^d\times\mathbb{R}^d\)上的分布时，必须找到一个平衡半空间以最小化切割对的比例。我们证明在无分布假设下，找到最优平衡半空间是NP难的。为规避此计算障碍，我们设计了一种高效的非恰当学习算法：若最优半空间切割\(\alpha\)比例的对，我们的算法输出一个度为\(\tilde{O}(1/\varepsilon^2)\)的平衡多项式阈值函数，其切割最多\(O(\sqrt{\alpha+\varepsilon})\)比例。
英文摘要
We study nearest neighbor search from the perspective of data-driven algorithm design: given a dataset $P \subset \mathbb{R}^d$ of size $n$ and sample access to a query distribution over $\mathbb{R}^d$, the goal is to learn a data structure optimized for queries drawn from that specific distribution. We focus on the class of balanced halfspace trees, which naturally abstracts space-partitioning frameworks like locality-sensitive hashing. Assuming Gaussian-like marginal conditions on the dataset and query distribution, we give an efficient algorithm that learns a tree achieving $o(nd)$ query time, provided that a perfect tree exists. At the core of our algorithmic approach is the balanced halfspace cut problem, where we are given a distribution over $\mathbb{R}^d \times \mathbb{R}^d$ and must find a balanced halfspace that minimizes the fraction of cut pairs. We prove that without distributional assumptions, finding the optimal balanced halfspace is NP-hard. To circumvent this computational barrier, we design an efficient improper learning algorithm: if the optimal halfspace cuts an $α$ fraction of pairs, our algorithm outputs a balanced polynomial threshold function of degree $\tilde{O}(1/\varepsilon^2)$ that cuts at most an $O(\sqrt{α+\varepsilon})$ fraction.
