CACM Article on Randomized Linear Algebra

Michael Mahoney

Each month the Communications of the ACM publishes an invited “Review Article” paper chosen from across the field of Computer Science.  These papers are intended to describe new developments of broad significance to the computing field, offer a high-level perspective on a technical area, and highlight unresolved questions and future directions.  The June 2016 issue of CACM contains a paper by AMPLab researcher Michael Mahoney and his colleague Petros Driness (of RPI, soon to be Purdue).  The paper, “RandNLA: Randomized Numerical Linear Algebra,” describes how randomization offers new benefits for large-scale linear algebra computations such as those that underlie a lot of the machine learning that is developed in the AMPLab and elsewhere.

Randomized Numerical Linear Algebra (RandNLA), a.k.a., Randomized Linear Algebra (RLA), is an interdisciplinary research area that exploits randomization as a computational resource to develop improved algorithms for large-scale linear algebra problems.  From a foundational perspective, RandNLA has its roots in theoretical computer science, with deep connections to mathematics (convex analysis, probability theory, metric embedding theory), applied mathematics (scientific computing, signal processing, numerical linear algebra), and theoretical statistics.  From an applied “big data” or “data science” perspective, RandNLA is a vital new tool for machine learning, statistics, and data analysis.  In addition, well-engineered implementations of RandNLA algorithms, e.g., Blendenpik, have already outperformed highly-optimized software libraries for ubiquitous problems such as least-squares.

Other RandNLA algorithms have good scalability in parallel and distributed environments.  In particular, AMPLab postdoc Alex Gittens has led an effort, in collaboration with researchers at Lawrence Berkeley National Laboratory and Cray, Inc., to explore the trade-offs of performing linear algebra computations such as RandNLA low-rank CX/CUR/PCA/NMF approximations at scale using Apache Spark, compared to traditional C and MPI implementations on HPC platforms, on LBNL’s supercomputers versus distributed data center computations.  As Alex describes in more detail in his recent AMPLab blog post, this project outlines Spark’s performance on some of the largest scientific data analysis workloads ever attempted with Spark, including using more than 48,000 cores on a supercomputing platform to compute the principal components of a 16TB atmospheric humidity data set.

Here are the key highlights from the CACM article on RandNLA.

1. Randomization isn’t just used to model noise in data; it can be a powerful computational resource to develop algorithms with improved running times and stability properties as well as algorithms that are more interpretable in downstream data science applications.

2. To achieve best results, random sampling of elements or columns/rows must be done carefully; but random projections can be used to transform or rotate the input data to a random basis where simple uniform random sampling of elements or rows/ columns can be successfully applied.

3. Random sketches can be used directly to get low-precision solutions to data science applications; or they can be used indirectly to construct preconditioners for traditional iterative numerical algorithms to get high-precision solutions in scientific computing applications.

More details on RandNLA can be found by clicking here for the full article and the associated video interview.