Open AccessPosted Content
Randomized Numerical Linear Algebra: Foundations & Algorithms.
TL;DR: This survey describes probabilistic algorithms for linear algebra computations, such as factorizing matrices and solving linear systems, that have a proven track record for real-world problem instances and treats both the theoretical foundations and the practical computational issues.
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Abstract: This survey describes probabilistic algorithms for linear algebra computations, such as factorizing matrices and solving linear systems. It focuses on techniques that have a proven track record for real-world problem instances. The paper treats both the theoretical foundations of the subject and the practical computational issues.
Topics covered include norm estimation; matrix approximation by sampling; structured and unstructured random embeddings; linear regression problems; low-rank approximation; subspace iteration and Krylov methods; error estimation and adaptivity; interpolatory and CUR factorizations; Nystrom approximation of positive-semidefinite matrices; single view ("streaming") algorithms; full rank-revealing factorizations; solvers for linear systems; and approximation of kernel matrices that arise in machine learning and in scientific computing.
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Citations
Randomized numerical linear algebra: Foundations and algorithms
TL;DR: This survey describes probabilistic algorithms for linear algebraic computations, such as factorizing matrices and solving linear systems, that have a proven track record for real-world problems and treats both the theoretical foundations of the subject and practical computational issues.
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.
Efficient Estimation of Pauli Observables by Derandomization
TL;DR: In this article, an efficient derandomization procedure that iteratively replaces random single-qubit measurements by fixed Pauli measurements was proposed, and the resulting deterministic measurement procedure is guaranteed to perform at least as well as the randomized one.
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Derivative-Informed Projected Neural Networks for High-Dimensional Parametric Maps Governed by PDEs
TL;DR: This work proposes to construct surrogates for high-dimensional PDE-governed parametric maps in the form of projected neural networks that parsimoniously capture the geometry and intrinsic low-dimensionality of these maps.
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Derivative-informed projected neural networks for high-dimensional parametric maps governed by PDEs
TL;DR: In this paper , a derivative-informed projected neural network (DIPNets) is proposed to construct surrogates for high-dimensional PDE-governed parametric maps, which parsimoniously capture the geometry and intrinsic low-dimensionality of these maps.
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Piotr Indyk,Rajeev Motwani +1 more
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TL;DR: In this paper, the authors present two algorithms for the approximate nearest neighbor problem in high-dimensional spaces, for data sets of size n living in R d, which require space that is only polynomial in n and d.