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 $\alpha$ fraction of pairs, our algorithm outputs a balanced polynomial threshold function of degree $\tilde{O}(1/\varepsilon^2)$ that cuts at most an $O(\sqrt{\alpha+\varepsilon})$ fraction.
Sanjeev Khanna, Ashwin Padaki, Erik Waingarten· 0 citations
We provide a local computation algorithm to approximate the top eigenvector $x \in \mathbb{R}^n$ of a symmetric matrix $A \in \mathbb{R}^{n \times n}$ with entries between $-1$ and $1$, building on the work of Swartworth and Woodruff [SODA 25] who show how to approximate the eigenvalues up to additive-$\varepsilon n$ error using $\tilde{O}(1/\varepsilon^4)$ queries. Our local computation algorithm has a preprocessing complexity of $\tilde{O}(1/\varepsilon^4)$ and per-coordinate query complexity of $\tilde{O}(1/\varepsilon^2)$ for an additive-$\varepsilon n$ approximation whenever {$|\lambda_{\min}(A)| = O(\lambda_{\max}(A))$. When $\lambda_{\min}(A)$ greatly exceeds $\lambda_{\max}(A)$, our complexity degrades to at most $\tilde{O}(1/\varepsilon^{6.\overline{6}})$ in preprocessing and $\tilde{O}(1/\varepsilon^{3.\overline{3}})$ per query. Furthermore, we show a lower bound of $\Omega(n/\varepsilon^2)$ on the total number of queries needed to output an approximately top eigenvector (implying that the per-coordinate query complexity of $\Omega(1/\varepsilon^2)$ is necessary). As an application, we use our algorithm to provide local computation algorithms for the sparsest-cut and max-cut problems in the dense graph model of Goldreich, Goldwasser, Ron [JACM 98]. By accessing the top eigenvectors (of an approximate normalized adjacency), we implement local versions of Cheeger's inequality and Trevisan's algorithm [SICOMP 12] to obtain"square-root-opt"approximations in polynomial time (as opposed to exponential-in-$\text{poly}(1/\varepsilon)$ time which is incurred in Goldreich, Goldwasser, Ron.
