Sketching as a Tool for Numerical Linear Algebra
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This survey highlights the recent advances in algorithms for numericallinear algebra that have come from the technique of linear sketching, and considers least squares as well as robust regression problems, low rank approximation, and graph sparsification.Abstract:
This survey highlights the recent advances in algorithms for numerical linear algebra that have come from the technique of linear sketching, whereby given a matrix, one first compresses it to a much smaller matrix by multiplying it by a (usually) random matrix with certain properties. Much of the expensive computation can then be performed on the smaller matrix, thereby accelerating the solution for the original problem. In this survey we consider least squares as well as robust regression problems, low rank approximation, and graph sparsification. We also discuss a number of variants of these problems. Finally, we discuss the limitations of sketching methods.read more
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Linear And Nonlinear Programming
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Newton-type methods for non-convex optimization under inexact Hessian information
TL;DR: In this article, the authors consider variants of trust-region and adaptive cubic regularization methods for non-convex optimization, in which the Hessian matrix is approximated, and provide iteration complexity to achieve $$\varepsilon $$ -approximate second-order optimality which have been shown to be tight.
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Practical sketching algorithms for low-rank matrix approximation
TL;DR: A suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image, or sketch, of the matrix that can preserve structural properties of the input matrix, such as positive-semidefiniteness, and they can produce approximation with a user-specified rank.
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Newton-Type Methods for Non-Convex Optimization Under Inexact Hessian Information
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Sub-sampled Newton Methods with Non-uniform Sampling
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