A Review for Weighted MinHash Algorithms <s> Algorithms <s> We describe an ecient procedure for sampling representatives from a weighted set such that the probability that for any weightings S and T, the probability that the two choose the same sample is the Jacard similarity: Pr[sample(S) = sample(T)] = P x min(S(x),T(x)) P x max(S(x),T(x)) . The sampling process takes expected time linear in the number of non-zero weights, independent of the weights themselves. We discuss and develop the implementation of our sampling schemes, reducing the requisite computation substantially, and reducing the randomness required to only four bits in expectation. <s> BIB001 </s> A Review for Weighted MinHash Algorithms <s> Algorithms <s> We propose a new Consistent Weighted Sampling method, where the probability of drawing identical samples for a pair of inputs is equal to their Jaccard similarity. Our method takes deterministic constant time per non-zero weight, improving on the best previous approach which takes expected constant time. The samples can be used as Weighted Minhash for efficient retrieval and compression (sketching) under Jaccard or L1 metric. A method is presented for using simple data statistics to reduce the running time of hash computation by two orders of magnitude. We compare our method with the random projection method and show that it has better characteristics for retrieval under L1. We present a novel method of mapping hashes to short bit-strings, apply it to Weighted Minhash, and achieve more accurate distance estimates from sketches than existing methods, as long as the inputs are sufficiently distinct. We show how to choose the optimal number of bits per hash for sketching, and demonstrate experimental results which agree with the theoretical analysis. <s> BIB002 </s> A Review for Weighted MinHash Algorithms <s> Algorithms <s> We develop 0-bit consistent weighted sampling (CWS) for efficiently estimating min-max kernel, which is a generalization of the resemblance kernel originally designed for binary data. Because the estimator of 0-bit CWS constitutes a positive definite kernel, this method can be naturally applied to large-scale data mining problems. Basically, if we feed the sampled data from 0-bit CWS to a highly efficient linear classifier (e.g., linear SVM), we effectively (and approximately) train a nonlinear classifier based on the min-max kernel. The accuracy improves as we increase the sample size. In this paper, we first demonstrate, through an extensive classification study using kernel machines, that the min-max kernel often provides an effective measure of similarity for nonnegative data. This helps justify the use of min-max kernel. However, as the min-max kernel is nonlinear and might be difficult to be used for industrial applications with massive data, we propose to linearize the min-max kernel via 0-bit CWS, a simplification of the original CWS method. The previous remarkable work on {\em consistent weighted sampling (CWS)} produces samples in the form of (i*, t*) where the i* records the location (and in fact also the weights) information analogous to the samples produced by classical minwise hashing on binary data. Because the t* is theoretically unbounded, it was not immediately clear how to effectively implement CWS for building large-scale linear classifiers. We provide a simple solution by discarding t* (which we refer to as the "0-bit" scheme). Via an extensive empirical study, we show that this 0-bit scheme does not lose essential information. We then apply 0-bit CWS for building linear classifiers to approximate min-max kernel classifiers, as extensively validated on a wide range of public datasets. We expect this work will generate interests among data mining practitioners who would like to efficiently utilize the nonlinear information of non-binary and nonnegative data. <s> BIB003 </s> A Review for Weighted MinHash Algorithms <s> Algorithms <s> Min-Hash, as a member of the Locality Sensitive Hashing (LSH) family for sketching sets, plays an important role in the big data era. It is widely used for efficiently estimating similarities of bag-of-words represented data and has been extended to dealing with multi-sets and real-value weighted sets. Improved Consistent Weighted Sampling (ICWS) has been recognized as the state-of-the-art for real-value weighted Min-Hash. However, the algorithmic implementation of ICWS is flawed because it violates the uniformity of the Min-Hash scheme. In this paper, we propose a Canonical Consistent Weighted Sampling (CCWS) algorithm, which not only retains the same theoretical complexity as ICWS but also strictly complies with the definition of Min-Hash. The experimental results demonstrate that the proposed CCWS algorithm runs faster than the state-of-the-arts while achieving similar classification performance on a number of real-world text data sets. <s> BIB004 </s> A Review for Weighted MinHash Algorithms <s> Algorithms <s> Min-Hash, which is widely used for efficiently estimating similarities of bag-of-words represented data, plays an increasingly important role in the era of big data. It has been extended to deal with real-value weighted sets -- Improved Consistent Weighted Sampling (ICWS) is considered as the state-of-the-art for this problem. In this paper, we propose a Practical CWS (PCWS) algorithm. We first transform the original form of ICWS into an equivalent expression, based on which we find some interesting properties that inspire us to make the ICWS algorithm simpler and more efficient in both space and time complexities. PCWS is not only mathematically equivalent to ICWS and preserves the same theoretical properties, but also saves 20% memory footprint and substantial computational cost compared to ICWS. The experimental results on a number of real-world text data sets demonstrate that PCWS obtains the same (even better) classification and retrieval performance as ICWS with 1/5~1/3 reduced empirical runtime. <s> BIB005
Brief Description CWS BIB001 Traverse several "active indices" ICWS BIB002 Sample two special "active indices" and generate hash code (k, y k ) 0-bit CWS BIB003 Discard y k produced by ICWS CCWS BIB004 Uniformly discretize the original weights instead of the logarithm of weighs in ICWS PCWS BIB005 Use one less number of uniform random variables than ICWS Relieve the dependence of two special "active indices" in ICWS