Relative Errors for Deterministic Low-Rank Matrix Approximations

We consider processing an n x d matrix A in a stream with row-wise updates according to a recent algorithm called Frequent Directions (Liberty, KDD 2013). This algorithm maintains an l x d matrix Q deterministically, processing each row in O(d l^2) time; the processing time can be decreased to O(d l) with a slight modification in the algorithm and a constant increase in space. We show that if one sets l = k+ k/eps and returns Q_k, a k x d matrix that is the best rank k approximation to Q, then we achieve the following properties: ||A - A_k||_F^2 <= ||A||_F^2 - ||Q_k||_F^2 <= (1+eps) ||A - A_k||_F^2 and where pi_{Q_k}(A) is the projection of A onto the rowspace of Q_k then ||A - pi_{Q_k}(A)||_F^2 <= (1+eps) ||A - A_k||_F^2. We also show that Frequent Directions cannot be adapted to a sparse version in an obvious way that retains the l original rows of the matrix, as opposed to a linear combination or sketch of the rows.Comment: 16 pages, 0 figure

Doctor of Philosophy

dissertationMatrices are essential data representations for many large-scale problems in data analytics; for example, in text analysis under the bag-of-words model, a large corpus of documents are often represented as a matrix. Many data analytic tasks rely on obtaining a summary (a.k.a sketch) of the data matrix. Using this summary in place of the original data matrix saves on space usage and run-time of machine learning algorithms. Therefore, sketching a matrix is often a necessary first step in data reduction, and sometimes has direct relationships to core techniques including PCA, LDA, and clustering. In this dissertation, we study the problem of matrix sketching over data streams. We first describe a deterministic matrix sketching algorithm called FrequentDirections. The algorithm is presented an arbitrary input matrix A∈ Rn&× d one row at a time. It performs O(dl) operations per row and maintains a sketch matrix B ∈ Rl× d such that for any k< l, ||ATA - BTB \|| 2 < ||A - Ak||F2 / (l-k) and ||A - πBk(A)||F2 ≤ (1 + k/l-k)||A-Ak||F2 . Here, Ak stands for the minimizer of ||A - Ak||F over all rank k matrices (similarly Bk), and πBk (A) is the rank k matrix resulting from projecting A on the row span of Bk. We show both of these bounds are the best possible for the space allowed, the sketch is mergeable, and hence trivially parallelizable. We propose several variants of FrequentDirections that improve its error-size tradeoff, and nearly matches the simple heuristic Iterative SVD method in practice. We then describe SparseFrequentDirections for sketching sparse matrices. It resembles the original algorithm in many ways including having the same optimal asymptotic guarantees with respect to the space-accuracy tradeoff in the streaming setting, but unlike FrequentDirections which runs in O(ndl) time, SparseFrequentDirections runs in Õ(nnz(A)l + nl2) time. We then extend our methods to distributed streaming model, where there are m distributed sites each observing a distinct stream of data, and which has a communication channel with a coordinator. The goal is to track an ε-approximation (for ε ∈ (0,1)) to the norm of the matrix along any direction. We present novel algorithms to address this problem. All our methods satisfy an additive error bound that for any unit vector x, | ||A x||2 - ||B x ||2 | ≤ |ε ||A||F2 holds

Improved Practical Matrix Sketching with Guarantees

Matrices have become essential data representations for many large-scale problems in data analytics, and hence matrix sketching is a critical task. Although much research has focused on improving the error/size tradeoff under various sketching paradigms, the many forms of error bounds make these approaches hard to compare in theory and in practice. This paper attempts to categorize and compare most known methods under row-wise streaming updates with provable guarantees, and then to tweak some of these methods to gain practical improvements while retaining guarantees. For instance, we observe that a simple heuristic iSVD, with no guarantees, tends to outperform all known approaches in terms of size/error trade-off. We modify the best performing method with guarantees FrequentDirections under the size/error trade-off to match the performance of iSVD and retain its guarantees. We also demonstrate some adversarial datasets where iSVD performs quite poorly. In comparing techniques in the time/error trade-off, techniques based on hashing or sampling tend to perform better. In this setting we modify the most studied sampling regime to retain error guarantee but obtain dramatic improvements in the time/error trade-off. Finally, we provide easy replication of our studies on APT, a new testbed which makes available not only code and datasets, but also a computing platform with fixed environmental settings.Comment: 27 page

Continuous Matrix Approximation on Distributed Data ∗

Tracking and approximating data matrices in streaming fashion is a fundamental challenge. The problem requires more care and attention when data comes from multiple distributed sites, each receiving a stream of data. This paper considers the problem of “tracking approximations to a matrix ” in the distributed streaming model. In this model, there are m distributed sites each observing a distinct stream of data (where each element is a row of a distributed matrix) and has a communication channel with a coordinator, and the goal is to track an ε-approximation to the norm of the matrix along any direction. To that end, we present novel algorithms to address the matrix approximation problem. Our algorithms maintain a smaller matrix B, as an approximation to a distributed streaming matrix A, such that for any unit vector x: |‖Ax ‖ 2 − ‖Bx ‖ 2 | ≤ ε‖A ‖ 2 F. Our algorithms work in streaming fashion and incur small communication, which is critical for distributed computation. Our best method is deterministic and uses only O((m/ε) log(βN)) communication, where N is the size of stream (at the time of the query) and β is an upperbound on the squared norm of any row of the matrix. In addition to proving all algorithmic properties theoretically, extensive experiments with real large datasets demonstrate the efficiency of these protocols. 1