Matrix Factorization 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 TB-sized problems in particle physics, climate modeling and bioimaging. The data matrices are tall-and-skinny which enable the algorithms to map conveniently into Spark's 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

ShapeFit and ShapeKick for Robust, Scalable Structure from Motion

We introduce a new method for location recovery from pair-wise directions that leverages an efficient convex program that comes with exact recovery guarantees, even in the presence of adversarial outliers. When pairwise directions represent scaled relative positions between pairs of views (estimated for instance with epipolar geometry) our method can be used for location recovery, that is the determination of relative pose up to a single unknown scale. For this task, our method yields performance comparable to the state-of-the-art with an order of magnitude speed-up. Our proposed numerical framework is flexible in that it accommodates other approaches to location recovery and can be used to speed up other methods. These properties are demonstrated by extensively testing against state-of-the-art methods for location recovery on 13 large, irregular collections of images of real scenes in addition to simulated data with ground truth

Scalable and distributed constrained low rank approximations

Low rank approximation is the problem of finding two low rank factors W and H such that the rank(WH) << rank(A) and A ≈ WH. These low rank factors W and H can be constrained for meaningful physical interpretation and referred as Constrained Low Rank Approximation (CLRA). Like most of the constrained optimization problem, performing CLRA can be computationally expensive than its unconstrained counterpart. A widely used CLRA is the Non-negative Matrix Factorization (NMF) which enforces non-negativity constraints in each of its low rank factors W and H. In this thesis, I focus on scalable/distributed CLRA algorithms for constraints such as boundedness and non-negativity for large real world matrices that includes text, High Definition (HD) video, social networks and recommender systems. First, I begin with the Bounded Matrix Low Rank Approximation (BMA) which imposes a lower and an upper bound on every element of the lower rank matrix. BMA is more challenging than NMF as it imposes bounds on the product WH rather than on each of the low rank factors W and H. For very large input matrices, we extend our BMA algorithm to Block BMA that can scale to a large number of processors. In applications, such as HD video, where the input matrix to be factored is extremely large, distributed computation is inevitable and the network communication becomes a major performance bottleneck. Towards this end, we propose a novel distributed Communication Avoiding NMF (CANMF) algorithm that communicates only the right low rank factor to its neighboring machine. Finally, a general distributed HPC- NMF framework that uses HPC techniques in communication intensive NMF operations and suitable for broader class of NMF algorithms.Ph.D

Fast and Guaranteed Tensor Decomposition via Sketching

Tensor CANDECOMP/PARAFAC (CP) decomposition has wide applications in statistical learning of latent variable models and in data mining. In this paper, we propose fast and randomized tensor CP decomposition algorithms based on sketching. We build on the idea of count sketches, but introduce many novel ideas which are unique to tensors. We develop novel methods for randomized computation of tensor contractions via FFTs, without explicitly forming the tensors. Such tensor contractions are encountered in decomposition methods such as tensor power iterations and alternating least squares. We also design novel colliding hashes for symmetric tensors to further save time in computing the sketches. We then combine these sketching ideas with existing whitening and tensor power iterative techniques to obtain the fastest algorithm on both sparse and dense tensors. The quality of approximation under our method does not depend on properties such as sparsity, uniformity of elements, etc. We apply the method for topic modeling and obtain competitive results.Comment: 29 pages. Appeared in Proceedings of Advances in Neural Information Processing Systems (NIPS), held at Montreal, Canada in 201