To Index or Not to Index: Optimizing Exact Maximum Inner Product Search

Exact Maximum Inner Product Search (MIPS) is an important task that is widely pertinent to recommender systems and high-dimensional similarity search. The brute-force approach to solving exact MIPS is computationally expensive, thus spurring recent development of novel indexes and pruning techniques for this task. In this paper, we show that a hardware-efficient brute-force approach, blocked matrix multiply (BMM), can outperform the state-of-the-art MIPS solvers by over an order of magnitude, for some -- but not all -- inputs. In this paper, we also present a novel MIPS solution, MAXIMUS, that takes advantage of hardware efficiency and pruning of the search space. Like BMM, MAXIMUS is faster than other solvers by up to an order of magnitude, but again only for some inputs. Since no single solution offers the best runtime performance for all inputs, we introduce a new data-dependent optimizer, OPTIMUS, that selects online with minimal overhead the best MIPS solver for a given input. Together, OPTIMUS and MAXIMUS outperform state-of-the-art MIPS solvers by 3.2$\times$ on average, and up to 10.9$\times$, on widely studied MIPS datasets.Comment: 12 pages, 8 figures, 2 table

Faster K-Means Cluster Estimation

There has been considerable work on improving popular clustering algorithm `K-means' in terms of mean squared error (MSE) and speed, both. However, most of the k-means variants tend to compute distance of each data point to each cluster centroid for every iteration. We propose a fast heuristic to overcome this bottleneck with only marginal increase in MSE. We observe that across all iterations of K-means, a data point changes its membership only among a small subset of clusters. Our heuristic predicts such clusters for each data point by looking at nearby clusters after the first iteration of k-means. We augment well known variants of k-means with our heuristic to demonstrate effectiveness of our heuristic. For various synthetic and real-world datasets, our heuristic achieves speed-up of up-to 3 times when compared to efficient variants of k-means.Comment: 6 pages, Accepted at ECIR 201

Faster k-Medoids Clustering: Improving the PAM, CLARA, and CLARANS Algorithms

Clustering non-Euclidean data is difficult, and one of the most used algorithms besides hierarchical clustering is the popular algorithm Partitioning Around Medoids (PAM), also simply referred to as k-medoids. In Euclidean geometry the mean-as used in k-means-is a good estimator for the cluster center, but this does not hold for arbitrary dissimilarities. PAM uses the medoid instead, the object with the smallest dissimilarity to all others in the cluster. This notion of centrality can be used with any (dis-)similarity, and thus is of high relevance to many domains such as biology that require the use of Jaccard, Gower, or more complex distances. A key issue with PAM is its high run time cost. We propose modifications to the PAM algorithm to achieve an O(k)-fold speedup in the second SWAP phase of the algorithm, but will still find the same results as the original PAM algorithm. If we slightly relax the choice of swaps performed (at comparable quality), we can further accelerate the algorithm by performing up to k swaps in each iteration. With the substantially faster SWAP, we can now also explore alternative strategies for choosing the initial medoids. We also show how the CLARA and CLARANS algorithms benefit from these modifications. It can easily be combined with earlier approaches to use PAM and CLARA on big data (some of which use PAM as a subroutine, hence can immediately benefit from these improvements), where the performance with high k becomes increasingly important. In experiments on real data with k=100, we observed a 200-fold speedup compared to the original PAM SWAP algorithm, making PAM applicable to larger data sets as long as we can afford to compute a distance matrix, and in particular to higher k (at k=2, the new SWAP was only 1.5 times faster, as the speedup is expected to increase with k)

Fast Approximate KK-Means via Cluster Closures

$K$-means, a simple and effective clustering algorithm, is one of the most widely used algorithms in multimedia and computer vision community. Traditional $k$-means is an iterative algorithm---in each iteration new cluster centers are computed and each data point is re-assigned to its nearest center. The cluster re-assignment step becomes prohibitively expensive when the number of data points and cluster centers are large. In this paper, we propose a novel approximate $k$-means algorithm to greatly reduce the computational complexity in the assignment step. Our approach is motivated by the observation that most active points changing their cluster assignments at each iteration are located on or near cluster boundaries. The idea is to efficiently identify those active points by pre-assembling the data into groups of neighboring points using multiple random spatial partition trees, and to use the neighborhood information to construct a closure for each cluster, in such a way only a small number of cluster candidates need to be considered when assigning a data point to its nearest cluster. Using complexity analysis, image data clustering, and applications to image retrieval, we show that our approach out-performs state-of-the-art approximate $k$-means algorithms in terms of clustering quality and efficiency

Fast k-means based on KNN Graph

In the era of big data, k-means clustering has been widely adopted as a basic processing tool in various contexts. However, its computational cost could be prohibitively high as the data size and the cluster number are large. It is well known that the processing bottleneck of k-means lies in the operation of seeking closest centroid in each iteration. In this paper, a novel solution towards the scalability issue of k-means is presented. In the proposal, k-means is supported by an approximate k-nearest neighbors graph. In the k-means iteration, each data sample is only compared to clusters that its nearest neighbors reside. Since the number of nearest neighbors we consider is much less than k, the processing cost in this step becomes minor and irrelevant to k. The processing bottleneck is therefore overcome. The most interesting thing is that k-nearest neighbor graph is constructed by iteratively calling the fast $k$-means itself. Comparing with existing fast k-means variants, the proposed algorithm achieves hundreds to thousands times speed-up while maintaining high clustering quality. As it is tested on 10 million 512-dimensional data, it takes only 5.2 hours to produce 1 million clusters. In contrast, to fulfill the same scale of clustering, it would take 3 years for traditional k-means

L-PICOLA: A parallel code for fast dark matter simulation

Robust measurements based on current large-scale structure surveys require precise knowledge of statistical and systematic errors. This can be obtained from large numbers of realistic mock galaxy catalogues that mimic the observed distribution of galaxies within the survey volume. To this end we present a fast, distributed-memory, planar-parallel code, L-PICOLA, which can be used to generate and evolve a set of initial conditions into a dark matter field much faster than a full non-linear N-Body simulation. Additionally, L-PICOLA has the ability to include primordial non-Gaussianity in the simulation and simulate the past lightcone at run-time, with optional replication of the simulation volume. Through comparisons to fully non-linear N-Body simulations we find that our code can reproduce the $z=0$ power spectrum and reduced bispectrum of dark matter to within 2% and 5% respectively on all scales of interest to measurements of Baryon Acoustic Oscillations and Redshift Space Distortions, but 3 orders of magnitude faster. The accuracy, speed and scalability of this code, alongside the additional features we have implemented, make it extremely useful for both current and next generation large-scale structure surveys. L-PICOLA is publicly available at https://cullanhowlett.github.io/l-picolaComment: 22 Pages, 20 Figures. Accepted for publication in Astronomy and Computin