Adaptive and Dynamic Multi-Resolution Hashing for Pairwise Summations

• URL: http://arxiv.org/abs/2212.11408v1
• Date: Wed, 21 Dec 2022 23:23:24 GMT
• Title: Adaptive and Dynamic Multi-Resolution Hashing for Pairwise Summations
• Authors: Lianke Qin, Aravind Reddy, Zhao Song, Zhaozhuo Xu, Danyang Zhuo
• Abstract summary: We propose Adam-Hash: an adaptive and dynamic multi-resolution hashing data-structure for fast pairwise summation estimation. Our proposed Adam-Hash is also robust to adaptive PSE queries, where an adversary can choose query $q_j in mathbbRd$ depending on the output from previous queries.
• Score: 19.602149096819776
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this paper, we propose Adam-Hash: an adaptive and dynamic multi-resolution hashing data-structure for fast pairwise summation estimation. Given a data-set $X \subset \mathbb{R}^d$, a binary function $f:\mathbb{R}^d\times \mathbb{R}^d\to \mathbb{R}$, and a point $y \in \mathbb{R}^d$, the Pairwise Summation Estimate $\mathrm{PSE}_X(y) := \frac{1}{|X|} \sum_{x \in X} f(x,y)$. For any given data-set $X$, we need to design a data-structure such that given any query point $y \in \mathbb{R}^d$, the data-structure approximately estimates $\mathrm{PSE}_X(y)$ in time that is sub-linear in $|X|$. Prior works on this problem have focused exclusively on the case where the data-set is static, and the queries are independent. In this paper, we design a hashing-based PSE data-structure which works for the more practical \textit{dynamic} setting in which insertions, deletions, and replacements of points are allowed. Moreover, our proposed Adam-Hash is also robust to adaptive PSE queries, where an adversary can choose query $q_j \in \mathbb{R}^d$ depending on the output from previous queries $q_1, q_2, \dots, q_{j-1}$.

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