## Making Very Large-Scale Linear Algebraic Computations Possible Via Randomization

author: Gunnar Martinsson, University of Colorado
published: Jan. 19, 2010,   recorded: December 2009,   views: 888
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# Slides

0:00 Slides Randomization: Making Very Large-Scale Linear Algebraic Computations Possible Goal The new methods -1 The new methods -2 The new methods -3 Applications -1 Caveats Applications -2 Diffusion Geometry -1 Diffusion Geometry -2 Diffusion Geometry -3 Example Picture -1 Picture -2 Solving the linear approximation problem Problem Question -1 Question -2 Question -3 Existing techniques for handling very large matrices Goal (restated) Related work Principal researchers on specific work reported How do you handle noise in the computation? -1 How do you handle noise in the computation? -2 Outline of the tutorial -1 Review of the SVD The decay of the singular values -1 The decay of the singular values -2 The SVD Model problem -1 Model problem -2 Model problem -3 Model problem -4 Model problem -5 Model problem -6 Model problem -7 Model problem -8 Picture -3 Golub-Businger algorithm Randomized method Theorem Proof Preview -1 Preview -2 Drawback Other randomized sampling strategies have been sugested Variations on the basic pattern Outline of the tutorial -2 A single pass algorithm for a symmetric matrix A single pass algorithm, continued -1 A single pass algorithm, continued -2 Outline of the tutorial -3 Finding "spanning columns" of a matrix -1 Finding "spanning columns" of a matrix -2 Finding "spanning columns" of a matrix -1 Finding "spanning columns" of a matrix -2 Finding "spanning columns" of a matrix -3 Finding "spanning columns" of a matrix -4 Finding "spanning columns" of a matrix -5 Picture -4 Picture -5 Picture -6 Picture -7 Picture -8 Picture -9 Picture -10 Picture -11 Picture -12 Factorization -1 Factorization -2

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# Description

The demands on software for analyzing data are rapidly increasing as ever larger data sets are generated in medical imaging, in analyzing large networks such as the World Wide Web, in image and video processing, and in a range of other applications. To handle this avalanche of data, any software used must be able to fully exploit modern hardware characterized by multiple processors and capacious but slow memory. The evolution of computer architecture is currently forcing a shift in algorithm design away from the classical algorithms that were designed for single-processor computers with all the data available in Random Access Memory (RAM), towards algorithms that aggressively minimize communication costs. This tutorial will describe a set of recently developed techniques for standard linear algebraic computations (such as computing a partial singular value decomposition of a matrix) that are very well suited for implementation on multi-core or other parallel architectures, and for processing data stored on disk, or streamed. These techniques are based on the use of randomized sampling to reduce the effective dimensionality of the data. Remarkably, randomized sampling does not only loosen communication constraints, but does so while maintaining, or even improving, the accuracy and robustness of existing deterministic techniques.