A Theory of Pseudoskeleton Approximations

By Sergei A. Goreinov, E. E. Tyrtyshnikov, Nickolai L. Zamarashkin. 1997

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A literature survey of low‐rank tensor approximation techniques (2013, neutral): “For an m × n matrix A , the so called cross approximation method [19, 34, 100, 248] produces an approximation of the form A (: , J ) A ( I, J ) − 1 A ( I,”
A randomized algorithm for a tensor-based generalization of the singular value decomposition (2007, background): “Goreinov, Tyrtyshnikov and Zamarashkin [22,21] were interested in applications such as scattering in which large coefﬁcient matrices have blocks that can be easily approximated by low-rank matrices.”
CUR matrix decompositions for improved data analysis (2009, background): “Similarly, Goreinov, Tyrtyshnikov, and Zamarashkin developed a CUR matrix decomposition (a pseudoskeleton approximation ) and related the choice of columns and rows to a “maximum uncorrelatedness” concept (9, 10).”
Fast Evaluation of Near-Field Boundary Integralsusing Tensor Approximations (2012, methodology): “12), a black box algorithm has been introduced in [41] which uses ideas from [22] for the approximation of the matricisations of a tensor.”
F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig Generalized Cross Approximation for 3d-tensors Generalized Cross Approximation for 3d-tensors (methodology): “It was introduced for d = 2 by Bebendorf in [1] as adaptive cross approximation (ACA), with forerunners by Tyrtyshnikov and others (see for instance [5]).”
Fast Direct Methods for Gaussian Processes (2014, methodology): “here is no a-priori information of the matrix, then algebraic methods provide an attractive way of computing fast low-rank decompositions. These include techniques like pseudo-skeletal approximations [23], interpolatory decomposition [17], randomized algorithms [19], [36], [52], rank-revealing LU [3”
Black box approximation of tensors in hierarchical Tucker format (2013, background): “1 Approximation of Matricizations In [4] the approximability of matrices by outer products of some of its columns and rows is analysed.”
Relative-Error CUR Matrix Decompositions (2007, background): “Goreinov, Tyrtyshnikov, and Zamarashkin [39, 38, 61] were interested in applications such as scattering, in which large coeﬃcient matrices have blocks that can be easily approximated by low-rank matrices.”
On the Nyström Method for Approximating a Gram Matrix for Improved Kernel-Based Learning (2005, background): “Examples may be found in [17], and also in [25, 24].”