Improved Densification of One Permutation Hashing

The existing work on densification of one permutation hashing reduces the query processing cost of the $(K,L)$-parameterized Locality Sensitive Hashing (LSH) algorithm with minwise hashing, from $O(dKL)$ to merely $O(d + KL)$, where $d$ is the number of nonzeros of the data vector, $K$ is the number of hashes in each hash table, and $L$ is the number of hash tables. While that is a substantial improvement, our analysis reveals that the existing densification scheme is sub-optimal. In particular, there is no enough randomness in that procedure, which affects its accuracy on very sparse datasets. In this paper, we provide a new densification procedure which is provably better than the existing scheme. This improvement is more significant for very sparse datasets which are common over the web. The improved technique has the same cost of $O(d + KL)$ for query processing, thereby making it strictly preferable over the existing procedure. Experimental evaluations on public datasets, in the task of hashing based near neighbor search, support our theoretical findings

Practical Hash Functions for Similarity Estimation and Dimensionality Reduction

Hashing is a basic tool for dimensionality reduction employed in several aspects of machine learning. However, the perfomance analysis is often carried out under the abstract assumption that a truly random unit cost hash function is used, without concern for which concrete hash function is employed. The concrete hash function may work fine on sufficiently random input. The question is if it can be trusted in the real world when faced with more structured input. In this paper we focus on two prominent applications of hashing, namely similarity estimation with the one permutation hashing (OPH) scheme of Li et al. [NIPS'12] and feature hashing (FH) of Weinberger et al. [ICML'09], both of which have found numerous applications, i.e. in approximate near-neighbour search with LSH and large-scale classification with SVM. We consider mixed tabulation hashing of Dahlgaard et al.[FOCS'15] which was proved to perform like a truly random hash function in many applications, including OPH. Here we first show improved concentration bounds for FH with truly random hashing and then argue that mixed tabulation performs similar for sparse input. Our main contribution, however, is an experimental comparison of different hashing schemes when used inside FH, OPH, and LSH. We find that mixed tabulation hashing is almost as fast as the multiply-mod-prime scheme ax+b mod p. Mutiply-mod-prime is guaranteed to work well on sufficiently random data, but we demonstrate that in the above applications, it can lead to bias and poor concentration on both real-world and synthetic data. We also compare with the popular MurmurHash3, which has no proven guarantees. Mixed tabulation and MurmurHash3 both perform similar to truly random hashing in our experiments. However, mixed tabulation is 40% faster than MurmurHash3, and it has the proven guarantee of good performance on all possible input.Comment: Preliminary version of this paper will appear at NIPS 201

Fast Similarity Sketching

We consider the Similarity Sketching problem: Given a universe $[u]= \{0,\ldots,u-1\}$ we want a random function $S$ mapping subsets $A\subseteq [u]$ into vectors $S(A)$ of size $t$, such that similarity is preserved. More precisely: Given sets $A,B\subseteq [u]$, define $X_i=[S(A)[i]= S(B)[i]]$ and $X=\sum_{i\in [t]}X_i$. We want to have $E[X]=t\cdot J(A,B)$, where $J(A,B)=|A\cap B|/|A\cup B|$ and furthermore to have strong concentration guarantees (i.e. Chernoff-style bounds) for $X$. This is a fundamental problem which has found numerous applications in data mining, large-scale classification, computer vision, similarity search, etc. via the classic MinHash algorithm. The vectors $S(A)$ are also called sketches. The seminal $t\times$MinHash algorithm uses $t$ random hash functions $h_1,\ldots, h_t$, and stores $\left(\min_{a\in A}h_1(A),\ldots, \min_{a\in A}h_t(A)\right)$ as the sketch of $A$. The main drawback of MinHash is, however, its $O(t\cdot |A|)$ running time, and finding a sketch with similar properties and faster running time has been the subject of several papers. Addressing this, Li et al. [NIPS'12] introduced one permutation hashing (OPH), which creates a sketch of size $t$ in $O(t + |A|)$ time, but with the drawback that possibly some of the $t$ entries are "empty" when $|A| = O(t)$. One could argue that sketching is not necessary in this case, however the desire in most applications is to have one sketching procedure that works for sets of all sizes. Therefore, filling out these empty entries is the subject of several follow-up papers initiated by Shrivastava and Li [ICML'14]. However, these "densification" schemes fail to provide good concentration bounds exactly in the case $|A| = O(t)$, where they are needed. (continued...

Differentially Private One Permutation Hashing and Bin-wise Consistent Weighted Sampling

Minwise hashing (MinHash) is a standard algorithm widely used in the industry, for large-scale search and learning applications with the binary (0/1) Jaccard similarity. One common use of MinHash is for processing massive n-gram text representations so that practitioners do not have to materialize the original data (which would be prohibitive). Another popular use of MinHash is for building hash tables to enable sub-linear time approximate near neighbor (ANN) search. MinHash has also been used as a tool for building large-scale machine learning systems. The standard implementation of MinHash requires applying $K$ random permutations. In comparison, the method of one permutation hashing (OPH), is an efficient alternative of MinHash which splits the data vectors into $K$ bins and generates hash values within each bin. OPH is substantially more efficient and also more convenient to use. In this paper, we combine the differential privacy (DP) with OPH (as well as MinHash), to propose the DP-OPH framework with three variants: DP-OPH-fix, DP-OPH-re and DP-OPH-rand, depending on which densification strategy is adopted to deal with empty bins in OPH. A detailed roadmap to the algorithm design is presented along with the privacy analysis. An analytical comparison of our proposed DP-OPH methods with the DP minwise hashing (DP-MH) is provided to justify the advantage of DP-OPH. Experiments on similarity search confirm the merits of DP-OPH, and guide the choice of the proper variant in different practical scenarios. Our technique is also extended to bin-wise consistent weighted sampling (BCWS) to develop a new DP algorithm called DP-BCWS for non-binary data. Experiments on classification tasks demonstrate that DP-BCWS is able to achieve excellent utility at around $\epsilon = 5\sim 10$, where $\epsilon$ is the standard parameter in the language of $(\epsilon, \delta)$-DP

A Memory-Efficient Sketch Method for Estimating High Similarities in Streaming Sets

Estimating set similarity and detecting highly similar sets are fundamental problems in areas such as databases, machine learning, and information retrieval. MinHash is a well-known technique for approximating Jaccard similarity of sets and has been successfully used for many applications such as similarity search and large scale learning. Its two compressed versions, b-bit MinHash and Odd Sketch, can significantly reduce the memory usage of the original MinHash method, especially for estimating high similarities (i.e., similarities around 1). Although MinHash can be applied to static sets as well as streaming sets, of which elements are given in a streaming fashion and cardinality is unknown or even infinite, unfortunately, b-bit MinHash and Odd Sketch fail to deal with streaming data. To solve this problem, we design a memory efficient sketch method, MaxLogHash, to accurately estimate Jaccard similarities in streaming sets. Compared to MinHash, our method uses smaller sized registers (each register consists of less than 7 bits) to build a compact sketch for each set. We also provide a simple yet accurate estimator for inferring Jaccard similarity from MaxLogHash sketches. In addition, we derive formulas for bounding the estimation error and determine the smallest necessary memory usage (i.e., the number of registers used for a MaxLogHash sketch) for the desired accuracy. We conduct experiments on a variety of datasets, and experimental results show that our method MaxLogHash is about 5 times more memory efficient than MinHash with the same accuracy and computational cost for estimating high similarities

Consistent Weighted Sampling Made Fast, Small, and Easy

Document sketching using Jaccard similarity has been a workable effective technique in reducing near-duplicates in Web page and image search results, and has also proven useful in file system synchronization, compression and learning applications. Min-wise sampling can be used to derive an unbiased estimator for Jaccard similarity and taking a few hundred independent consistent samples leads to compact sketches which provide good estimates of pairwise-similarity. Subsequent works extended this technique to weighted sets and show how to produce samples with only a constant number of hash evaluations for any element, independent of its weight. Another improvement by Li et al. shows how to speedup sketch computations by computing many (near-)independent samples in one shot. Unfortunately this latter improvement works only for the unweighted case. In this paper we give a simple, fast and accurate procedure which reduces weighted sets to unweighted sets with small impact on the Jaccard similarity. This leads to compact sketches consisting of many (near-)independent weighted samples which can be computed with just a small constant number of hash function evaluations per weighted element. The size of the produced unweighted set is furthermore a tunable parameter which enables us to run the unweighted scheme of Li et al. in the regime where it is most efficient. Even when the sets involved are unweighted, our approach gives a simple solution to the densification problem that other works attempted to address. Unlike previously known schemes, ours does not result in an unbiased estimator. However, we prove that the bias introduced by our reduction is negligible and that the standard deviation is comparable to the unweighted case. We also empirically evaluate our scheme and show that it gives significant gains in computational efficiency, without any measurable loss in accuracy