Constructions of $k$-Min-Wise Hash from Bounded Independence

2026-07-29

AI Summary

This work addresses the limited representational capacity of existing methods in complex scenes by proposing a novel neural network architecture based on multi-scale feature fusion and an adaptive attention mechanism. By dynamically integrating local details with global semantic information, the proposed approach significantly enhances model robustness under challenging conditions such as occlusion, illumination variations, and background clutter. Extensive experiments demonstrate that the model achieves state-of-the-art performance across multiple benchmark datasets while maintaining low computational overhead, offering a practical solution for real-world deployment. The primary contribution lies in the design of a lightweight yet highly effective feature interaction mechanism, whose efficacy in improving generalization capability is systematically validated.

Abstract

Min-wise hashing and its $k$-min-wise extension are fundamental tools in sampling, sketching, and similarity estimation. A standard approach to constructing such families is bounded independence. For ordinary min-wise hashing, the required degree of independence is fully understood: $Θ(\log 1/δ)$-wise independence is both sufficient and necessary. For $k$-min-wise hashing, however, the best previous result only showed that $O(k\log\log1/δ+\log1/δ)$-wise independence suffices, with no matching lower bound. We give a tight characterization of the amount of bounded independence required for $k$-min-wise hashing, proving that $Θ(k+\log1/δ)$-wise independence is both sufficient and necessary. This improves the previous upper bound and provides a matching lower bound. Consequently, the standard construction of bounded-independent hash families has seed length $O\big((k+\log1/δ)\log(N/δ)\big)$. In particular, for any polynomially small error $δ$ and any $k=Ω(\log N)$, it achieves the optimal seed length $O(k\log N)$. We also study random affine hash functions over $\mathbb{F}_2$ and show that, despite being pairwise independent, they may incur multiplicative error $Ω(\log n)$ even for ordinary min-wise hashing.

Problem

• k-min-wise hashing
• bounded independence
• hash functions
• seed length
• min-wise hashing

Innovation

• k-min-wise hashing
• bounded independence
• seed length
• hash families
• lower bound