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Fast algorithms for spectral collocation with non-periodic boundary conditions
2005
Journal of Computational Physics
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The underlying ideas are introduced and developed in the context of linearly implicit methods for stiff equations. ...
By employing a structured representation of linear operators we are able to use fast algorithms without being restricted to periodic boundary conditions. ...
The off-diagonal blocks in this partitioning are shown to have low rank. ...
doi:10.1016/j.jcp.2005.01.007
fatcat:xefok3i3fvfdtis4enyqfzxms4
Eigenvalue computation for unitary rank structured matrices
2008
Journal of Computational and Applied Mathematics
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In this paper we describe how to compute the eigenvalues of a unitary rank structured matrix in two steps. ...
First we perform a reduction of the given matrix into Hessenberg form, next we compute the eigenvalues of this resulting Hessenberg matrix via an implicit QR-algorithm. ...
low rank blocks to the main diagonal, and where n denotes the matrix size. ...
doi:10.1016/j.cam.2007.01.006
fatcat:ibaihjs22rhfveeac4ylwg7wca
Block computation and representation of a sparse nullspace basis of a rectangular matrix
2008
Linear Algebra and its Applications
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We will employ this observation to design an efficient block algorithm that computes a sparse representation of the nullspace basis in almost optimal complexity. ...
We assume that B has full rank, i.e., rank(B) = m. It is well known that the last n − m columns of the orthogonal matrix Q in a QR factorization B = QR form such a desired null basis. ...
Efficient realization of the implicit block representation by ordering and low rank approximation In this section, we propose some strategies which will greatly decrease the computational costs of computing ...
doi:10.1016/j.laa.2007.11.025
fatcat:ykzck3flqrgxnldjii2by6kq5q
Fast QR iterations for unitary plus low rank matrices
[article]
2019
arXiv
pre-print
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2018 pre-print arXiv:1810.02708v1 fatcat:g6a7nqh32ravzf2nepcz4wur24
Some fast algorithms for computing the eigenvalues of a block companion matrix A = U + XY^H, where U∈C^n× n is unitary block circulant and X, Y ∈C^n × k, have recently appeared in the literature. ...
A remarkable case is U unitary diagonal which makes possible to deal with interpolation techniques for rootfinding problems and nonlinear eigenvalue problems. ...
Matrix Q (1) is still block-diagonal with the leading block k × k unitary diagonal, and the tailing block Hessenberg. ...
arXiv:1810.02708v2
fatcat:oykg7yfnzfclvkata4m5nv7coe
hm-toolbox: Matlab software for HODLR and HSS matrices
[article]
2020
arXiv
pre-print
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Matrices with hierarchical low-rank structure, including HODLR and HSS matrices, constitute a versatile tool to develop fast algorithms for addressing large-scale problems. ...
Nevertheless, it maintains the favorable complexity of hierarchical low-rank matrices and offers, at the same time, a convenient way of prototyping and experimenting with algorithms. ...
An HSS matrix is converted into a HODLR matrix by simply building explicit low-rank factorizations of the off-diagonal blocks from their implicit nested representation in the HSS format. ...
arXiv:1909.07909v3
fatcat:blvtvqamh5dxlaktli3fpugvoe
Understanding Dimensional Collapse in Contrastive Self-supervised Learning
[article]
2022
arXiv
pre-print
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Self-supervised visual representation learning aims to learn useful representations without relying on human annotations. ...
Inspired by our theory, we propose a novel contrastive learning method, called DirectCLR, which directly optimizes the representation space without relying on an explicit trainable projector. ...
Projector diagonal low-rank Top-1 Accuracy no projector 51.5 orthogonal projector 52.2 trainable projector 61.1 trainable diagonal projector 60.2 fixed low-rank projector 62.3 fixed low-rank diagonal projector ...
arXiv:2110.09348v3
fatcat:3etdtubetvcl5blmcrl24bscuy
Efficient Reduction of Compressed Unitary plus Low-rank Matrices to Hessenberg form
[article]
2019
arXiv
pre-print
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We present fast numerical methods for computing the Hessenberg reduction of a unitary plus low-rank matrix A=G+U V^H, where G∈C^n× n is a unitary matrix represented in some compressed format using O(nk ...
Small Rank Modifications of Unitary Block Diagonal Matrices. ...
In particular, the representation is suited for the fast eigensolver for unitary plus low rank matrices developed in [8] . Our derivation is based on three key ingredients or building blocks: 1. ...
arXiv:1901.08411v2
fatcat:5p6pwym6u5cj3ogmuzdengbmci
Efficient construction of an HSS preconditioner for symmetric positive definite ℋ^2 matrices
[article]
2021
arXiv
pre-print
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Fast (linear-scaling) matrix-vector products are available by expressing the kernel matrix in an ℋ^2 representation or an equivalent fast multipole method representation. ...
Preconditioning such matrices, however, requires a structured matrix approximation that is more regular than the ℋ^2 representation, such as the hierarchically semiseparable (HSS) matrix representation ...
The HSS representation generally requires more of its off-diagonal blocks to be compressed into low-rank form than the H 2 representation. ...
arXiv:2011.07632v2
fatcat:w2l7e45cvzeitj7lg7rmwpzjuu
Implicit double shift QR-algorithm for companion matrices
2010
Numerische Mathematik
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Moreover, no compression is needed as the specific representation of the involved matrices is maintained. Finally, also a double shift version of the implicit method is presented. ...
Moreover, no compression is needed as the specific representation of the involved matrices is maintained. Finally, also a double shift version of the implicit method is presented. ...
The representation designed in this section is the one that will be used for developing an implicit QR-method for unitary plus low rank matrices. ...
doi:10.1007/s00211-010-0302-y
fatcat:boutu3uz4vhvvowfkwvdcoty3y
Unitary rank structured matrices
2008
Journal of Computational and Applied Mathematics
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pull-through process of the two branches of the representation. ...
We also provide some basic operations for manipulating the representation, such as the transition to zero-creating form, the transition to a unitary/Givens-weight representation, as well as an internal ...
It reveals then that for each pure structure block of a unitary matrix, the complementary submatrix must have low rank as well. ...
doi:10.1016/j.cam.2007.03.020
fatcat:jmgli3tg6ngufiej3jtjrrx4mi
A bibliography on semiseparable matrices*
2005
Calcolo
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matrices, while in other cases they were defined as matrices having low rank blocks below the diagonal. ...
matrices, while in other cases they were defined as matrices having low rank blocks below the diagonal. ...
-One can generalize all the above representations to come to block forms. Such as recursively semiseparable, sequentially semiseparable matrices or low Hankel rank matrices. ...
doi:10.1007/s10092-005-0107-z
fatcat:dj3ema4whveg3l725gofhrpir4
Fast algorithms for hierarchically semiseparable matrices
2010
Numerical Linear Algebra with Applications
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Moreover, we provide a new linear complexity explicit U LV factorization algorithm for symmetric positive definite HSS matrices with a low-rank property. ...
In this paper, we generalize the hierarchically semiseparable (HSS) matrix representations and propose some fast algorithms for HSS matrices. ...
The low-rank property is concerned with the ranks or numerical ranks of certain types of off-diagonal blocks. ...
doi:10.1002/nla.691
fatcat:neof3jzipfdhxevk53ha5jwb2u
A preconditioned GMRES for complex dense linear systems from electromagnetic wave scattering problems
2006
Linear Algebra and its Applications
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The novelty of this work is the construction of an approximate hierarchically semiseparable (HSS) representation of the near-field matrix, the part of the matrix capturing interactions among nearby groups ...
In the light of this experience we propose a multilevel near-field matrix and its corresponding HSS representation as a hierarchical preconditioner in order to substantially reduce the number of iterations ...
HSS matrices are characterized by a hierarchical low-rank structure in the off-diagonal blocks. ...
doi:10.1016/j.laa.2005.11.022
fatcat:nbnnnabtfndn7fgoakcqzoboci
Efficient Reduction of Compressed Unitary Plus Low Rank Matrices to Hessenberg Form
2020
SIAM Journal on Matrix Analysis and Applications
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We present fast numerical methods for computing the Hessenberg reduction of a unitary plus low rank matrix A = G + U V H , where G \in \BbbC n\times n is a unitary matrix represented in some compressed ...
Small rank modifications of unitary block diagonal matrices. ...
However the Hessenberg matrix also obtained with that algorithm, which is based on the Givens-vector representation of the low rank part, is not directly exploitable by available fast eigensolvers. ...
doi:10.1137/19m1280363
fatcat:6wnrhmeuqzc2hmxwvg6orizbn4
Discriminative Block-Diagonal Representation Learning for Image Recognition
2018
IEEE Transactions on Neural Networks and Learning Systems
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In this paper, we propose a discriminative block-diagonal low-rank representation (BDLRR) method for recognition. ...
representation under the semi-supervised framework of low-rank representation. ...
For CBDS, it locally enforces the classwise diagonal structure on the low-rank criterion, whereas our BDLRR method globally imposes the block-diagonal constraint on the low-rank criterion by directly minimizing ...
doi:10.1109/tnnls.2017.2712801
pmid:28692990
fatcat:53nnhax2tvh4finsqw2tevxaaq
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