Factorizing large matrices by QR with column pivoting (QRCP) is substantially more expensive than QR without pivoting, owing to communication costs required for pivoting decisions. In contrast, randomized QRCP (RQRCP) algorithms have proven themselves empirically to be highly competitive with high-performance implementations of QR in processing time, on uniprocessor and shared memory machines, and as reliable as QRCP in pivot quality. We show that RQRCP algorithms can be as reliable as QRCP with failure probabilities exponentially decaying in oversampling size. We also analyze efficiency differences among different RQRCP algorithms. More importantly, we develop distributed memory implementations of RQRCP that are significantly better than QRCP implementations in ScaLAPACK. As a further development, we introduce the concept of and develop algorithms for computing spectrum-revealing QR factorizations for low-rank matrix approximations, and demonstrate their effectiveness against leading low-rank approximation methods in both theoretical and numerical reliability and efficiency.

Low-rank matrix approximation is a fundamental tool in data analysis for processing large datasets, reducing noise, and finding important signals. In this work, we present a novel truncated LU factorization called Spectrum-Revealing LU (SRLU) for effective low-rank matrix approximation, and develop a fast algorithm to compute an SRLU factorization. We provide both matrix and singular value approximation error bounds for the SRLU approximation computed by our algorithm. Our analysis suggests that SRLU is competitive with the best low-rank matrix approximation methods, deterministic or randomized, in both computational complexity and approximation quality. Numeric experiments illustrate that SRLU preserves sparsity, highlights important data features and variables, can be efficiently updated, and calculates data approximations nearly as accurately as the best possible. To the best of our knowledge this is the first practical variant of the LU factorization for effective and efficient low-rank matrix approximation.

Kernel methods represent some of the most popular machine learning tools for data analysis. Since exact kernel methods can be prohibitively expensive for large problems, reliable low-rank matrix approximations and high-performance implementations have become indispensable for practical applications of kernel methods. In this work, we introduce spectrum-revealing Cholesky factorization, a reliable low-rank matrix factorization, for kernel matrix approximation. We also develop an efficient and effective randomized algorithm for computing this factorization. Our numerical experiments demonstrate that this algorithm is as effective as other Cholesky factorization based kernel methods on machine learning problems, but significantly faster.

The CUR matrix decomposition and the related Nyström method build low-rank approximations of data matrices by selecting a small number of representative rows and columns of the data. Here, we introduce novel \emphspectral gap error bounds that judiciously exploit the potentially rapid spectrum decay in the input matrix, a most common occurrence in machine learning and data analysis. Our error bounds are much tighter than existing ones for matrices with rapid spectrum decay, and they justify the use of a constant amount of oversampling relative to the rank parameter k, i.e, when the number of columns/rows is \ell=k+ O(1). We demonstrate our analysis on a novel deterministic algorithm, \emphStableCUR, which additionally eliminates a previously unrecognized source of potential instability in CUR decompositions. While our algorithm accepts any method of row and column selection, we implement it with a recent column selection scheme with strong singular value bounds. Empirical results on various classes of real world data matrices demonstrate that our algorithm is as efficient as and often outperforms competing algorithms.

Electricity is a vital part of our daily life; therefore it is important to avoid irregularities such as the California Electricity Crisis of 2000 and 2001. In this work, we seek to predict anomalies using advanced machine learning algorithms, more specifically a Change Point Detection (CPD) algorithm on the electricity prices during the California Electricity Crisis. Such algorithms are effective, but computationally expensive when applied on a large amount of data. To address this challenge, we accelerate the Gaussian Process (GP) for 1-dimensional time series data. Since GP is at the core of many statistical learning techniques, this improvement could benefit many algorithms. In the specific Change Point Detection algorithm used in this study, we reduce the overall computational complexity from O(n5) to O(n2), where the amountized cost of solving a GP projet is O(1). Our efficient algorithm makes it possible to compute the Change Points using the hourly price data during the California Electricity Crisis. By comparing the detected Change Points with known events, we show that the Change Point Detection algorithm is indeed effective in detecting signals preceding major events.

Many data types arising from data mining applications can be modeled as bipartite graphs, examples include terms and documents in a text corpus, customers and purchasing items in market basket analysis and reviewers and movies in a movie recommender system. In this paper, we propose a new data clustering method based on partitioning the underlying bipartite graph. The partition is constructed by minimizing a normalized sum of edge weights between unmatched pairs of vertices of the bipartite graph. We show that an approximate solution to the minimization problem can be obtained by computing a partial singular value decomposition (SVD) of the associated edge weight matrix of the bipartite graph. We point out the connection of our clustering algorithm to correspondence analysis used in multivariate analysis. We also briefly discuss the issue of assigning data objects to multiple clusters. In the experimental results, we apply our clustering algorithm to the problem of document clustering to illustrate its effectiveness and efficiency.

An important application of graph partitioning is data clustering using a graph model - the pairwise similarities between all data objects form a weighted graph adjacency matrix that contains all necessary information for clustering. In this paper, we propose a new algorithm for graph partitioning with an objective function that follows the min-max clustering principle. The relaxed version of the optimization of the min-max cut objective function leads to the Fiedler vector in spectral graph partitioning. Theoretical analyses of min-max cut indicate that it leads to balanced partitions, and lower bounds are derived. The min-max cut algorithm is tested on newsgroup data sets and is found to out-perform other current popular partitioning/clustering methods. The linkage-based refinements to the algorithm further improve the quality of clustering substantially. We also demonstrate that a linearized search order based on linkage differential is better than that based on the Fiedler vector, providing another effective partitioning method.

The popular K-means clustering partitions a data set by minimiz(cid:173) ing a sum-of-squares cost function. A coordinate descend method is then used to find local minima. In this paper we show that the minimization can be reformulated as a trace maximization problem associated with the Gram matrix of the data vectors. Furthermore, we show that a relaxed version of the trace maximization problem possesses global optimal solutions which can be obtained by com(cid:173) puting a partial eigendecomposition of the Gram matrix, and the cluster assignment for each data vectors can be found by comput(cid:173) ing a pivoted QR decomposition of the eigenvector matrix. As a by-product we also derive a lower bound for the minimum of the sum-of-squares cost function.

We formulate and solve a new parameter estimation problem in the presence of bounded data uncertainties. The new method is suitable when a priori bounds on the uncertain data are available; its solution guarantees that the effect of the uncertainties will never be unnecessarily overestimated beyond what is reasonably assumed by the a priori bounds.