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

Dimensionality Reduction for k-Means Clustering and Low Rank Approximation

We show how to approximate a data matrix $\mathbf{A}$ with a much smaller sketch $\mathbf{\tilde A}$ that can be used to solve a general class of constrained k-rank approximation problems to within $(1+\epsilon)$ error. Importantly, this class of problems includes $k$-means clustering and unconstrained low rank approximation (i.e. principal component analysis). By reducing data points to just $O(k)$ dimensions, our methods generically accelerate any exact, approximate, or heuristic algorithm for these ubiquitous problems. For $k$-means dimensionality reduction, we provide $(1+\epsilon)$ relative error results for many common sketching techniques, including random row projection, column selection, and approximate SVD. For approximate principal component analysis, we give a simple alternative to known algorithms that has applications in the streaming setting. Additionally, we extend recent work on column-based matrix reconstruction, giving column subsets that not only `cover' a good subspace for \bv{A}, but can be used directly to compute this subspace. Finally, for $k$-means clustering, we show how to achieve a $(9+\epsilon)$ approximation by Johnson-Lindenstrauss projecting data points to just $O(\log k/\epsilon^2)$ dimensions. This gives the first result that leverages the specific structure of $k$-means to achieve dimension independent of input size and sublinear in $k$

Optimal Principal Component Analysis in Distributed and Streaming Models

We study the Principal Component Analysis (PCA) problem in the distributed and streaming models of computation. Given a matrix $A \in R^{m \times n},$ a rank parameter $k < rank(A)$, and an accuracy parameter $0 < \epsilon < 1$, we want to output an $m \times k$ orthonormal matrix $U$ for which $$|| A - U U^T A ||_F^2 \le \left(1 + \epsilon \right) \cdot || A - A_k||_F^2,$$ where $A_k \in R^{m \times n}$ is the best rank-$k$ approximation to $A$. This paper provides improved algorithms for distributed PCA and streaming PCA.Comment: STOC2016 full versio