Alex Gittens

Improved matrix algorithms via the Subsampled Randomized Hadamard Transform

Several recent randomized linear algebra algorithms rely upon fast dimension reduction methods. A popular choice is the subsampled randomized Hadamard transform (SRHT). In this article, we address the efficacy, in the Frobenius and spectral norms, of an SRHT-based low-rank matrix approximation technique introduced by Woolfe, Liberty, Rohklin, and Tygert. We establish a slightly better Frobenius norm error bound than is currently available, and a much sharper spectral norm error bound (in the presence of reasonable decay of the singular values). Along the way, we produce several results on matrix operations with SRHTs (such as approximate matrix multiplication) that may be of independent interest. Our approach builds upon Tropps in “Improved Analysis of the Subsampled Randomized Hadamard Transform” [Adv. Adaptive Data Anal., 3 (2011), pp. 115--126].

Matrix factorizations at scale: A comparison of scientific data analytics in spark and C+MPI using three case studies

We explore the trade-offs of performing linear algebra using Apache Spark, compared to traditional C and MPI implementations on HPC platforms. Spark is designed for data analytics on cluster computing platforms with access to local disks and is optimized for data-parallel tasks. We examine three widely-used and important matrix factorizations: NMF (for physical plausability), PCA (for its ubiquity) and CX (for data interpretability). We apply these methods to 1.6TB particle physics, 2.2TB and 16TB climate modeling and 1.1TB bioimaging data. The data matrices are tall-and-skinny which enable the algorithms to map conveniently into Sparks data-parallel model. We perform scaling experiments on up to 1600 Cray XC40 nodes, describe the sources of slowdowns, and provide tuning guidance to obtain high performance.

Skip-Gram - Zipf + Uniform = Vector Additivity

In recent years word-embedding models have gained great popularity due to their remarkable performance on several tasks, including word analogy questions and caption generation. An unexpected “side-effect” of such models is that their vectors often exhibit compositionality, i.e., addingtwo word-vectors results in a vector that is only a small angle away from the vector of a word representing the semantic composite of the original words, e.g., “man” + “royal” = “king”. This work provides a theoretical justification for the presence of additive compositionality in word vectors learned using the Skip-Gram model. In particular, it shows that additive compositionality holds in an even stricter sense (small distance rather than small angle) under certain assumptions on the process generating the corpus. As a corollary, it explains the success of vector calculus in solving word analogies. When these assumptions do not hold, this work describes the correct non-linear composition operator. Finally, this work establishes a connection between the Skip-Gram model and the Sufficient Dimensionality Reduction (SDR) framework of Globerson and Tishby: the parameters of SDR models can be obtained from those of Skip-Gram models simply by adding information on symbol frequencies. This shows that Skip-Gram embeddings are optimal in the sense of Globerson and Tishby and, further, implies that the heuristics commonly used to approximately fit Skip-Gram models can be used to fit SDR models.

A Multi-Platform Evaluation of the Randomized CX Low-Rank Matrix Factorization in Spark

We investigate the performance and scalability of the randomized CX low-rank matrix factorization and demonstrate its applicability through the analysis of a 1TB mass spectrometry imaging (MSI) dataset, using Apache Spark on an Amazon EC2 cluster, a Cray XC40 system, and an experimental Cray cluster. We implemented this factorization both as a parallelized C implementation with hand-tuned optimizations and in Scala using the Apache Spark high-level cluster computing framework. We obtained consistent performance across the three platforms: using Spark we were able to process the 1TB size dataset in under 30 minutes with 960 cores on all systems, with the fastest times obtained on the experimental Cray cluster. In comparison, the C implementation processed the 1TB size dataset 21X faster on the Amazon EC2 system, due to careful cache optimizations, bandwidth-friendly access of matrices and vector computation using SIMD units. We report these results and their implications on the hardware and software issues arising in supporting data-centric workloads in parallel and distributed environments.

Hardware compliant approximate image codes

In recent years, several feature encoding schemes for the bags-of-visual-words model have been proposed. While most of these schemes produce impressive results, they all share an important limitation: their high computational complexity makes it challenging to use them for large-scale problems. In this work, we propose an approximate locality-constrained encoding scheme that offers significantly better computational efficiency (~ 40×) than its exact counterpart, with comparable classification accuracy. Using the perturbation analysis of least-squares problems, we present a formal approximation error analysis of our approach, which helps distill the intuition behind the robustness of our method. We present a thorough set of empirical analyses on multiple standard data-sets, to assess the capability of our encoding scheme for its representational as well as discriminative accuracy.