Maximum Inner Product Search (MIPS) is a popular task to find the vector with the highest inner product for a given query. ScaNN is a score-aware quantization approach for MIPS that effectively transforms vectors with higher inner products into short sequences of codewords within codebooks. When quantizing vectors, it iteratively updates codebooks by assigning vectors to codewords and computing inverse matrices obtained from the assigned vectors. ScaNN, however, incurs a high computation cost when quantizing large-scale data. This is because (1) it computes quantization losses for all pairs of vectors and codewords, and (2) the size of the inverse matrices is quadratic in the number of dimensions. Our proposal, F-ScaNN, increases the efficiency of ScaNN through two techniques: (1) it computes the upper and lower bounds of the losses to assign vectors, and (2) it employs the conjugate gradient method to avoid computing the inverse matrix. Theoretically, we can obtain the same quantization results as ScaNN. Furthermore, we can improve search accuracy by using scaled codewords. Experiments show that our approach is significantly faster than previous approaches.

K-Multiple-Means is an extension of K-means for the clustering of multiple means used in many applications, such as image segmentation, load balancing, and blind-source separation. Since K-means uses only one mean to represent each cluster, it fails to capture non-spherical cluster structures of data points. However, since K-Multiple-Means represents the cluster by computing multiple means and grouping them into specified c clusters, it can effectively capture the non-spherical clusters of the data points. To obtain the clusters, K-Multiple-Means updates a similarity matrix of a bipartite graph between the data points and the multiple means by iteratively computing the leading c singular vectors of the matrix. K-Multiple-Means, however, incurs a high computation cost for large-scale data due to the iterative SVD computations. Our proposal, F-KMM, increases the efficiency of K-Multiple-Means by computing the singular vectors from a smaller similarity matrix between the multiple means obtained from the similarity matrix of the bipartite graph. To compute the similarity matrix of the bipartite graph efficiently, we skip unnecessary distance computations and estimate lower bounding distances between the data points and the multiple means. Theoretically, the proposed approach guarantees the same clustering results as K-Multiple-Means since it can exactly compute the singular vectors from the similarity matrix between the multiple means. Experiments show that our approach is several orders of magnitude faster than previous clustering approaches that use multiple means.

Abstract Network representation learning is a de facto tool for graph analytics. The mainstream of the previous approaches is to factorize the proximity matrix between nodes. However, if n is the number of nodes, since the size of the proximity matrix is $$n \times n$$ n × n , it needs $$O(n^3)$$ O ( n 3 ) time and $$O(n^2)$$ O ( n 2 ) space to perform network representation learning; they are significantly high for large-scale graphs. This paper introduces the novel idea of using similarities between clusters instead of proximities between nodes; the proposed approach computes the representations of the clusters from similarities between clusters and computes the representations of nodes by referring to them. If l is the number of clusters, since $$l \ll n$$ l ≪ n , we can efficiently obtain the representations of clusters from a small $$l \times l$$ l × l similarity matrix. Furthermore, since nodes in each cluster share similar structural properties, we can effectively compute the representation vectors of nodes. Experiments show that our approach can perform network representation learning more efficiently and effectively than existing approaches.

Network hashing converts each node of a graph into a compact binary code, and it is a useful graph analytics tool since it can reduce memory cost. INH-MF is a network hashing approach to factorize the high-order proximity matrix representing similarities between nodes. However, since it cuts small nonzero elements from the proximity matrix, it fails to effectively extract insights from the graph. Moreover, it incurs high memory and computational costs since the proximity matrix is large and dense. We propose Graph Clustering-based Network Hashing, a novel network hashing approach. To compute the proximities effectively, it uses the structural relationships between nodes and clusters obtained from a graph clustering approach. Moreover, it can efficiently compute hash codes from eigenvectors of the matrix corresponding to the graph Laplacian by using its low-rank property. Experiments show that it can more efficiently and effectively compute hash codes than previous approaches.

Anchor graphs are a popular tool used in label prediction of sparsely labeled data. In anchor graphs, labels of labeled data are propagated to unlabeled data via anchor points; anchor points are the centers of k-means clusters. Anchor graph-based label prediction determines local weights between data points and anchor points by exploiting Nesterov's method to obtain the graph's adjacency matrix, and it inverts a matrix obtained from the adjacency matrix to predict labels., however, incurs high computation cost since (1) Nesterov's method is applied to all closest anchor points to compute local weights, and (2) the computation cost of the inversion matrix is cubic in the number of anchor points. We propose an approach that can efficiently perform anchor graph-based label prediction because of its two key advances: (1) it prunes unnecessary anchor points so they are not passed to Nesterov's method, and (2) it applies the conjugate gradient method in computing labels of data points to avoid matrix inversion. In addition, we propose to exploit basis vectors computed by SVD as anchor points to improve label prediction accuracy. Experiments show that our approach outperforms the previous approaches in terms of efficiency and accuracy.

Data visualization has become a fundamental process of data engineering. t-SNE is one of the most popular data visualization approaches. However, its computation cost is quadratic to the number of data points because it needs to compute similarities for all pairs of data points. One practical way of using t-SNE is random walk-based t-SNE. This approach visualizes user-specified landmark points from the similarities between them based on random walks in a neighborhood graph of data points. It offers two approaches to computing similarities: the direct and analytical approaches. The direct approach approximately computes similarities by explicitly computing random walks in the graph. Unfortunately, it needs to perform numerous random walks for adequate computation accuracy. The analytical approach performs Cholesky factorization on the graph Laplacian and computes exact similarities using the decomposed graph Laplacian. This, however, incurs high computation cost in performing Cholesky factorization. Our proposal, F-tSNE, reduces the computation cost of random walk-based t-SNE by computing the LDL decomposition for the graph Laplacian based on two ideas: (1) reducing non-zero elements in the LDL decomposition by using a reordering matrix and (2) exploiting the sparse structure of the graph when computing the similarities. Theoretically, our approach is guaranteed to yield exact similarities. Experiments show that it is up to 88.4 times faster than the existing alternatives.

Anchor graph hashing is used in many applications such as cancer detection, web page classification, and drug discovery. It computes the hash codes from the eigenvectors of the matrix representing the similarities between data points and anchor points; anchors refer to the points representing the data distribution. In performing an approximate nearest neighbor search, the hash codes of a query data point are determined by identifying its closest anchor points. Anchor graph hashing, however, incurs high computation cost since (1) the computation cost of obtaining the eigenvectors is quadratic to the number of anchor points, and (2) the similarities of the query data point to all the anchor points must be computed. Our proposal, Tridiagonal hashing , increases the efficiency of anchor graph hashing because of its two advances: (1) we apply a graph clustering algorithm to compute the eigenvectors from the tridiagonal matrix obtained from the similarities between data points and anchor points, and (2) we detect anchor points closest to the query data point by using a dimensionality reduction approach. Experiments show that our approach is several orders of magnitude faster than the previous approaches. Besides, it yields high search accuracy than the original anchor graph hashing approach.

The b-matching graph is a useful approach to computing a graph from high-dimensional data. Unlike the k-NN graph that greedily connects each data point to its k nearest neighbors and typically has more than k edges, each data point in the b-matching graph uniformly has b edges; the idea is reduce edges between cross-clusters that have different semantics. In addition, edge weights are obtained from regression results of each data pointand restricted to be non-negative to improve the robustness for data noise. The b-matching graph can more effectively model high-dimensional data than the traditional k-NN graph. However, the construction cost of the b-matching graph is impractical for large-scale data sets. This is because, to determine edges in the graph, it needs to iteratively update messages between all pairs of data points until convergence, and it computes non-negative edge weights of each data point by applying a solver intended for quadratic programming problems. Our proposal, b-dash, can efficiently construct a b-matching graph because of its two key techniques: (1) it prunes unnecessary update messages in determining edges and (2) it incrementally computes edge weights by exploiting the Sherman-Morrison formula. Experiments show that our approach is up to 58.6 times faster than the previous approaches while guaranteeing result optimality.

Random forest is an ensemble approach based on decision trees. It computes the best split in each node in terms of impurity reduction. However, the impurity computations incur high computation cost in its training process. This paper proposes F-forest, an efficient variant of random forest. It incrementally estimates upper bounds for scores that correspond to impurity reductions to find the best split. Since we can safely skip unnecessary computations, it can guarantee the same training result as the original approach. Experiments show that our approach is faster than state-of-the-art approaches.

Support Vector Machine (SVM) is one of the most popular classification algorithms. SVM separates data points into two classes by using the hyper-plane that is maximally distant from the two classes. Since SVM is theoretically based on statistical learning theory and the principle of structural risk minimization, it offers highly accurate classification. However, its training process is computationally expensive. This paper proposes Sahara as an efficient training algorithm for SVM. It identifies data points that have no influence on SVM classification by computing the upper and lower bounds of a parameter that determines the hyper-plane. Our approach can efficiently compute the bounds by using Singular Value Decomposition (SVD) and a sparse data matrix. Theoretically, our approach guarantees to yield the optimal hyper-plane of SVM for any given set of data points. Experiments show that Sahara is significantly faster than previous approaches.

The lasso-based L 1 -graph is used in many applications since it can effectively model a set of data points as a graph. The lasso is a popular regression approach and the L 1 -graph represents data points as nodes by using the regression result. More specifically, by solving the L 1 -optimization problem of the lasso, the sparse regression coefficients are used to obtain the weights of the edges in the graph. Conventional graph structures such as k -NN graph use two steps, adjacency searching and weight selection, for constructing the graph whereas the lasso-based L 1 -graph derives the adjacency structure as well as the edge weights simultaneously by using a coordinate descent. However, the construction cost of the lasso-based L 1 -graph is impractical for large data sets since the coordinate descent iteratively updates the weights of all edges until convergence. Our proposal, Castnet, can efficiently construct the lasso-based L 1 -graph. In order to avoid updating the weights of all edges, we prune edges that cannot have nonzero weights before entering the iterations. In addition, we update edge weights only if they are nonzero in the iterations. Experiments show that Castnet is significantly faster than existing approaches.

Affinity Propagation is a clustering algorithm used in many applications. It iteratively updates messages between data points until convergence. The message updating process enables Affinity Propagation to have higher clustering quality compared with other approaches. However, its computation cost is high; it is quadratic in the number of data points. This is because it updates the messages of all data point pairs. This paper proposes an efficient algorithm that guarantees the same clustering results as the original algorithm. Our approach, F-AP, is based on two ideas: (1) it computes upper and lower estimates to limit the messages to be updated in each iteration, and (2) it dynamically detects converged messages to efficiently skip unneeded updates. Experiments show that F-AP is much faster than previous approaches with no loss in clustering performance.

We propose a domain-dependent/independent topic switching model based on Bayesian probabilistic modeling for modeling online product reviews that are accompanied with numerical ratings provided by users. In this model, each word is allocated to a domain-dependent topic or a domain-independent topic, and the distribution of topics in an online review is connected to an observed numerical rating via a linear regression model. Domain-dependent topics utilize domain information observed with a corpus, and domain-independent topics utilize the framework of Bayesian Nonparametrics, which can estimate the number of topics in posterior distributions. The posterior distribution is estimated via collapsed Gibbs sampling. Using real data, our proposed model had smaller mean square error and smaller average mean error with a small model size and achieved convergence in fewer iterations for a regression task involving online review ratings, outperforming a baseline model that did not consider domains. Moreover, the proposed model can also tell us whether the words are positive or negative in the form of continuous values. This feature allows us to extract domain-dependent and -independent sentiment words.