In response to the need for learning tools tuned to big data analytics, the present paper introduces a framework for efficient clustering of huge sets of (possibly high-dimensional) data. Building on random sampling and consensus (RANSAC) ideas pursued earlier in a different (computer vision) context for robust regression, a suite of novel dimensionality and set-reduction algorithms is developed. The advocated sketch-and-validate (SkeVa) family includes two algorithms that rely on K-means clustering per iteration on reduced number of dimensions and/or feature vectors: The first operates in a batch fashion, while the second sequential one offers computational efficiency and suitability with streaming modes of operation. For clustering even nonlinearly separable vectors, the SkeVa family offers also a member based on user-selected kernel functions. Further trading off performance for reduced complexity, a fourth member of the SkeVa family is based on a divergence criterion for selecting proper minimal subsets of feature variables and vectors, thus bypassing the need for K-means clustering per iteration. Extensive numerical tests on synthetic and real data sets highlight the potential of the proposed algorithms, and demonstrate their competitive performance relative to state-of-the-art random projection alternatives.

While computer and communication technologies have provided effective means to scale up many aspects of education, the submission and grading of assessments such as homework assignments and tests remains a weak link. In this paper, we study the problem of automatically grading the kinds of open response mathematical questions that figure prominently in STEM (science, technology, engineering, and mathematics) courses. Our data-driven framework for mathematical language processing (MLP) leverages solution data from a large number of learners to evaluate the correctness of their solutions, assign partial-credit scores, and provide feedback to each learner on the likely locations of any errors. MLP takes inspiration from the success of natural language processing for text data and comprises three main steps. First, we convert each solution to an open response mathematical question into a series of numerical features. Second, we cluster the features from several solutions to uncover the structures of correct, partially correct, and incorrect solutions. We develop two different clustering approaches, one that leverages generic clustering algorithms and one based on Bayesian nonparametrics. Third, we automatically grade the remaining (potentially large number of) solutions based on their assigned cluster and one instructor-provided grade per cluster. As a bonus, we can track the cluster assignment of each step of a multistep solution and determine when it departs from a cluster of correct solutions, which enables us to indicate the likely locations of errors to learners. We test and validate MLP on real-world MOOC data to demonstrate how it can substantially reduce the human effort required in large-scale educational platforms.

A natural approach to analyze interaction data of form "what-connects-to-what-when" is to create a time-series (or rather a sequence) of graphs through temporal discretization (bandwidth selection) and spatial discretization (vertex contraction). Such discretization together with non-negative factorization techniques can be useful for obtaining clustering of graphs. Motivating application of performing clustering of graphs (as opposed to vertex clustering) can be found in neuroscience and in social network analysis, and it can also be used to enhance community detection (i.e., vertex clustering) by way of conditioning on the cluster labels. In this paper, we formulate a problem of clustering of graphs as a model selection problem. Our approach involves information criteria, non-negative matrix factorization and singular value thresholding, and we illustrate our techniques using real and simulated data.

Directed graphs have asymmetric connections, yet the current graph clustering methodologies cannot identify the potentially global structure of these asymmetries. We give a spectral algorithm called di-sim that builds on a dual measure of similarity that correspond to how a node (i) sends and (ii) receives edges. Using di-sim, we analyze the global asymmetries in the networks of Enron emails, political blogs, and the c elegans neural connectome. In each example, a small subset of nodes have persistent asymmetries; these nodes send edges with one cluster, but receive edges with another cluster. Previous approaches would have assigned these asymmetric nodes to only one cluster, failing to identify their sending/receiving asymmetries. Regularization and "projection" are two steps of di-sim that are essential for spectral clustering algorithms to work in practice. The theoretical results show that these steps make the algorithm weakly consistent under the degree corrected Stochastic co-Blockmodel, a model that generalizes the Stochastic Blockmodel to allow for both (i) degree heterogeneity and (ii) the global asymmetries that we intend to detect. The theoretical results make no assumptions on the smallest degree nodes. Instead, the theorem requires that the average degree grows sufficiently fast and that the weak consistency only applies to the subset of the nodes with sufficiently large leverage scores. The results results also apply to bipartite graphs.

Diverse experimental data ranging from microarray gene expression data in biology to spectrum data in astronomy require to be clustered to signal meaningful correlation of the data. Massive documents or images on internet are also needed to be effectively organized so as to promote the efficiency of search engines. Clustering method as K-means (1) is popular for its simplicity, yet sensitive to noise and initialization and thus is limited by the lack of reliability. Hierarchical clustering (HC) (2) is simple and intuitive and thus widely used especially in biology (3), whereas it needs a large computation (4) and its result is variable to a set of similarity measures between clusters. Moreover, the cluster number for the above methods needs to be prespecified (e.g., K-means) or determined by a threshold (e.g., HC). Some other well-known algorithms either involve complex optimization and postprocessing (5), or have limited range of applications such as the distribution (6) or the attribute of data (7, 8). Although affinity propagation (AP) (9) has much better performance than K-means and the cluster number is determined automatically, it is not good at detecting nonspherical clusters (10). Recently, two effective clustering algorithms (10, 11) were proposed, which can together form a pool of clustering methods based on the in-tree structure (11). But they involve a free parameter.

High-dimensional, simultaneous recordings of neural spiking activity are often explored, analyzed and visualized with the help of latent variable or factor models. Such models are however ill-equipped to extract structure beyond shared, distributed aspects of firing activity across multiple cells. Here, we extend unstructured factor models by proposing a model that discovers subpopulations or groups of cells from the pool of recorded neurons. The model combines aspects of mixture of factor analyzer models for capturing clustering structure, and aspects of latent dynamical system models for capturing temporal dependencies. In the resulting model, we infer the subpopulations and the latent factors from data using variational inference and model parameters are estimated by Expectation Maximization (EM). We also address the crucial problem of initializing parameters for EM by extending a sparse subspace clustering algorithm to integer-valued spike count observations. We illustrate the merits of the proposed model by applying it to calcium-imaging data from spinal cord neurons, and we show that it uncovers meaningful clustering structure in the data.

In this paper, we consider sparse networks consisting of a finite number of non-overlapping communities, i.e. disjoint clusters, so that there is higher density within clusters than across clusters. Both the intra- and inter-cluster edge densities vanish when the size of the graph grows large, making the cluster reconstruction problem nosier and hence difficult to solve. We are interested in scenarios where the network size is very large, so that the adjacency matrix of the graph is hard to manipulate and store. The data stream model in which columns of the adjacency matrix are revealed sequentially constitutes a natural framework in this setting. For this model, we develop two novel clustering algorithms that extract the clusters asymptotically accurately. The first algorithm is {\it offline}, as it needs to store and keep the assignments of nodes to clusters, and requires a memory that scales linearly with the network size. The second algorithm is {\it online}, as it may classify a node when the corresponding column is revealed and then discard this information. This algorithm requires a memory growing sub-linearly with the network size. To construct these efficient streaming memory-limited clustering algorithms, we first address the problem of clustering with partial information, where only a small proportion of the columns of the adjacency matrix is observed and develop, for this setting, a new spectral algorithm which is of independent interest.

Spectral Clustering as a relaxation of the normalized/ratio cut has become one of the standard graph-based clustering methods. Existing methods for the computation of multiple clusters, corresponding to a balanced k-cut of the graph, are either based on greedy techniques or heuristics which have weak connection to the original motivation of minimizing the normalized cut. In this paper we propose a new tight continuous relaxation for any balanced k-cut problem and show that a related recently proposed relaxation is in most cases loose leading to poor performance in practice. For the optimization of our tight continuous relaxation we propose a new algorithm for the hard sum-of-ratios minimization problem which achieves monotonic descent. Extensive comparisons show that our method beats all existing approaches for ratio cut and other balanced k-cut criteria.

We consider the problem of finding clusters in an unweighted graph, when the graph is partially observed. We analyze two programs, one which works for dense graphs and one which works for both sparse and dense graphs, but requires some a priori knowledge of the total cluster size, that are based on the convex optimization approach for low-rank matrix recovery using nuclear norm minimization. For the commonly used Stochastic Block Model, we obtain \emph{explicit} bounds on the parameters of the problem (size and sparsity of clusters, the amount of observed data) and the regularization parameter characterize the success and failure of the programs. We corroborate our theoretical findings through extensive simulations. We also run our algorithm on a real data set obtained from crowdsourcing an image classification task on the Amazon Mechanical Turk, and observe significant performance improvement over traditional methods such as k-means.

Computing the $k$ dominant eigenvalues and eigenvectors of massive graphs is a key operation in numerous machine learning applications; however, popular solvers suffer from slow convergence, especially when $k$ is reasonably large. In this paper, we propose and analyze a novel multi-scale spectral decomposition method (MSEIGS), which first clusters the graph into smaller clusters whose spectral decomposition can be computed efficiently and independently. We show theoretically as well as empirically that the union of all cluster's subspaces has significant overlap with the dominant subspace of the original graph, provided that the graph is clustered appropriately. Thus, eigenvectors of the clusters serve as good initializations to a block Lanczos algorithm that is used to compute spectral decomposition of the original graph. We further use hierarchical clustering to speed up the computation and adopt a fast early termination strategy to compute quality approximations. Our method outperforms widely used solvers in terms of convergence speed and approximation quality. Furthermore, our method is naturally parallelizable and exhibits significant speedups in shared-memory parallel settings. For example, on a graph with more than 82 million nodes and 3.6 billion edges, MSEIGS takes less than 3 hours on a single-core machine while Randomized SVD takes more than 6 hours, to obtain a similar approximation of the top-50 eigenvectors. Using 16 cores, we can reduce this time to less than 40 minutes.