On Densification for Minwise Hashing
Tung Mai, Anup Rao, Matt Kapilevich, Ryan Rossi, Yasin Abbasi-Yadkori, Ritwik Sinha
Abstract
One Permutation Hashing (OPH) is a significantly more efficient alternative to the popular minwise hashing. To produce a sketch of size $k$, OPH requires just one hash function whereas the classical minwise hashing requires $k$ hash functions. However, OPH does not have the desirable locality sensitive hashing (LSH) property that is important for indexing. [Srivastava and Li 2014, ICML] proposed the novel idea of densification to produce LSH sketches from OPH sketches, and gave the first densification routine. In this paper, we give a necessary and sufficient condition for a densification routine to result in LSH sketches when applied to OPH sketches. Furthermore, we give a novel densification routine that for every input, takes O($k \log k$) time in expectation and achieves better variance than the previous best bound obtained by [Srivastava 2017]. The running time of the densification routine given in [Srivastava 2017] for worst case inputs is O($k^2$) in expectation.
BibTeX
@InProceedings{pmlr-v115-mai20a,
title = {On Densification for Minwise Hashing},
author = {Mai, Tung and Rao, Anup and Kapilevich, Matt and Rossi, Ryan and Abbasi-Yadkori, Yasin and Sinha, Ritwik},
booktitle = {Proceedings of The 35th Uncertainty in Artificial Intelligence Conference},
pages = {831--840},
year = {2020},
editor = {Adams, Ryan P. and Gogate, Vibhav},
volume = {115},
series = {Proceedings of Machine Learning Research},
month = {22--25 Jul},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v115/mai20a/mai20a.pdf},
url = {https://proceedings.mlr.press/v115/mai20a.html},
abstract = {One Permutation Hashing (OPH) is a significantly more efficient alternative to the popular minwise hashing. To produce a sketch of size $k$, OPH requires just one hash function whereas the classical minwise hashing requires $k$ hash functions. However, OPH does not have the desirable locality sensitive hashing (LSH) property that is important for indexing. [Srivastava and Li 2014, ICML] proposed the novel idea of densification to produce LSH sketches from OPH sketches, and gave the first densification routine. In this paper, we give a necessary and sufficient condition for a densification routine to result in LSH sketches when applied to OPH sketches. Furthermore, we give a novel densification routine that for every input, takes O($k \log k$) time in expectation and achieves better variance than the previous best bound obtained by [Srivastava 2017]. The running time of the densification routine given in [Srivastava 2017] for worst case inputs is O($k^2$) in expectation.}
}