ﻻ يوجد ملخص باللغة العربية
Traditional minwise hashing (MinHash) requires applying $K$ independent permutations to estimate the Jaccard similarity in massive binary (0/1) data, where $K$ can be (e.g.,) 1024 or even larger, depending on applications. The recent work on C-MinHash (Li and Li, 2021) has shown, with rigorous proofs, that only two permutations are needed. An initial permutation is applied to break whatever structures which might exist in the data, and a second permutation is re-used $K$ times to produce $K$ hashes, via a circulant shifting fashion. (Li and Li, 2021) has proved that, perhaps surprisingly, even though the $K$ hashes are correlated, the estimation variance is strictly smaller than the variance of the traditional MinHash. It has been demonstrated in (Li and Li, 2021) that the initial permutation in C-MinHash is indeed necessary. For the ease of theoretical analysis, they have used two independent permutations. In this paper, we show that one can actually simply use one permutation. That is, one single permutation is used for both the initial pre-processing step to break the structures in the data and the circulant hashing step to generate $K$ hashes. Although the theoretical analysis becomes very complicated, we are able to explicitly write down the expression for the expectation of the estimator. The new estimator is no longer unbiased but the bias is extremely small and has essentially no impact on the estimation accuracy (mean square errors). An extensive set of experiments are provided to verify our claim for using just one permutation.
Minwise hashing (MinHash) is an important and practical algorithm for generating random hashes to approximate the Jaccard (resemblance) similarity in massive binary (0/1) data. The basic theory of MinHash requires applying hundreds or even thousands
In this extended abstract, we describe and analyze a lossy compression of MinHash from buckets of size $O(log n)$ to buckets of size $O(loglog n)$ by encoding using floating-point notation. This new compressed sketch, which we call HyperMinHash, as w
Given a Counting Monadic Second Order (CMSO) sentence $psi$, the CMSO$[psi]$ problem is defined as follows. The input to CMSO$[psi]$ is a graph $G$, and the objective is to determine whether $Gmodels psi$. Our main theorem states that for every CMSO
We initiate the study of property testing problems concerning relations between permutations. In such problems, the input is a tuple $(sigma_1,dotsc,sigma_d)$ of permutations on ${1,dotsc,n}$, and one wishes to determine whether this tuple satisfies
The complexity of a computational problem is traditionally quantified based on the hardness of its worst case. This approach has many advantages and has led to a deep and beautiful theory. However, from the practical perspective, this leaves much to