We consider the problem of sparse matrix multiplication by the column row method in a distributed setting where the matrix product is not necessarily sparse. We present a surprisingly simple method for "consistent" parallel processing of sparse outer products (column-row vector products) over several processors, in a communication-avoiding setting where each processor has a copy of the input. The method is consistent in the sense that a given output entry is always assigned to the same processor independently of the specific structure of the outer product. We show guarantees on the work done by each processor, and achieve linear speedup down to the point where the cost is dominated by reading the input. Our method gives a way of distributing (or parallelizing) matrix product computations in settings where the main bottlenecks are storing the result matrix, and inter-processor communication. Motivated by observations on real data that often the absolute values of the entries in the product adhere to a power law, we combine our approach with frequent items mining algorithms and show how to obtain a tight approximation of the weight of the heaviest entries in the product matrix. As a case study we present the application of our approach to frequent pair mining in transactional data streams, a problem that can be phrased in terms of sparse @math -integer matrix multiplication by the column-row method. Experimental evaluation of the proposed method on real-life data supports the theoretical findings.
Manku and Motwani @cite_6 first recognized the necessity for efficient algorithms targeted at frequent itemsets in transaction streams and presented a heuristic approach generalizing their StickySampling algorithm. A straightforward approach to mining of frequent pairs is to reduce the problem to that of mining frequent items by generating all item pairs in a given transaction. @cite_9 and Campagna and Pagh @cite_7 present randomized algorithms for transaction stream mining. The theoretical bounds on the quality of their estimates however heavily depend on the assumption that transactions are either generated independently at random by some process or arrive in a random order. It is already clear from the experiments of @cite_7 that such optimistic assumptions do not hold for many data sets. For both schemes @cite_9 @cite_7 it is easy to find an ordering of essentially any transaction stream that breaks the randomness assumption, and makes it perform much worse than the theoretical bounds. We therefore believe that a more conservative model is needed to derive a rigorous theoretical analysis, while exploiting observed properties of real data sets.
@cite_9 @cite_7 @cite_6
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Mining frequent itemsets from transactional data streams is challenging due to the nature of the exponential explosion of itemsets and the limit memory space required for mining frequent itemsets. Given a domain of I unique items, the possible number of itemsets can be up to 2^I-1. When the length of data streams approaches to a very large number N, the possibility of an itemset to be frequent becomes larger and difficult to track with limited memory. The existing studies on finding frequent items from high speed data streams are false-positive oriented. That is, they control memory consumption in the counting processes by an error parameter @e, and allow items with support below the specified minimum support s but above [email protected] counted as frequent ones. However, such false-positive oriented approaches cannot be effectively applied to frequent itemsets mining for two reasons. First, false-positive items found increase the number of false-positive frequent itemsets exponentially. Second, minimization of the number of false-positive items found, by using a small @e, will make memory consumption large. Therefore, such approaches may make the problem computationally intractable with bounded memory consumption. In this paper, we developed algorithms that can effectively mine frequent item(set)s from high speed transactional data streams with a bound of memory consumption. Our algorithms are based on Chernoff bound in which we use a running error parameter to prune item(set)s and use a reliability parameter to control memory. While our algorithms are false-negative oriented, that is, certain frequent itemsets may not appear in the results, the number of false-negative itemsets can be controlled by a predefined parameter so that desired recall rate of frequent itemsets can be guaranteed. Our extensive experimental studies show that the proposed algorithms have high accuracy, require less memory, and consume less CPU time. They significantly outperform the existing false-positive algorithms. In this paper, we examine the issue of mining association rules among items in a large database of sales transactions. The mining of association rules can be mapped into the problem of discovering large itemsets where a large itemset is a group of items which appear in a sufficient number of transactions. The problem of discovering large itemsets can be solved by constructing a candidate set of itemsets first and then, identifying, within this candidate set, those itemsets that meet the large itemset requirement. Generally this is done iteratively for each large k-itemset in increasing order of k where a large k-itemset is a large itemset with k items. To determine large itemsets from a huge number of candidate large itemsets in early iterations is usually the dominating factor for the overall data mining performance. To address this issue, we propose an effective hash-based algorithm for the candidate set generation. Explicitly, the number of candidate 2-itemsets generated by the proposed algorithm is, in orders of magnitude, smaller than that by previous methods, thus resolving the performance bottleneck. Note that the generation of smaller candidate sets enables us to effectively trim the transaction database size at a much earlier stage of the iterations, thereby reducing the computational cost for later iterations significantly. Extensive simulation study is conducted to evaluate performance of the proposed algorithm. The problem of assessing the significance of data mining results on high-dimensional 0-1 data sets has been studied extensively in the literature. For problems such as mining frequent sets and finding correlations, significance testing can be done by, e.g., chi-square tests, or many other methods. However, the results of such tests depend only on the specific attributes and not on the dataset as a whole. Moreover, the tests are more difficult to apply to sets of patterns or other complex results of data mining. In this paper, we consider a simple randomization technique that deals with this shortcoming. The approach consists of producing random datasets that have the same row and column margins with the given dataset, computing the results of interest on the randomized instances, and comparing them against the results on the actual data. This randomization technique can be used to assess the results of many different types of data mining algorithms, such as frequent sets, clustering, and rankings. To generate random datasets with given margins, we use variations of a Markov chain approach, which is based on a simple swap operation. We give theoretical results on the efficiency of different randomization methods, and apply the swap randomization method to several well-known datasets. Our results indicate that for some datasets the structure discovered by the data mining algorithms is a random artifact, while for other datasets the discovered structure conveys meaningful information. We study the problem of mining frequent itemsets from uncertain data under a probabilistic framework. We consider transactions whose items are associated with existential probabilities and give a formal definition of frequent patterns under such an uncertain data model. We show that traditional algorithms for mining frequent itemsets are either inapplicable or computationally inefficient under such a model. A data trimming framework is proposed to improve mining efficiency. Through extensive experiments, we show that the data trimming technique can achieve significant savings in both CPU cost and I O cost.