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Matrix rigidity of random toeplitz matrices

Oded Goldreich, Avishay Tal
2016 Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2016  
We prove that random n-by-n Toeplitz (alternatively Hankel) matrices over F 2 have rigidity Ω( n 3 r 2 log n ) for rank r ≥ √ n, with high probability.  ...  This improves, for r = o(n/ log n log log n), over the Ω( n 2 r · log( n r )) bound that is known for many explicit matrices.  ...  Theorem 1.3 (on the structured rigidity of random Toeplitz/Hankel matrices). Let A ∈ F n×n 2 be a random Toeplitz/Hankel matrix.  ... 
doi:10.1145/2897518.2897633 dblp:conf/stoc/GoldreichT16 fatcat:xhkbzqr7mjf6be6embtsvnqd5e

Matrix rigidity of random Toeplitz matrices

Oded Goldreich, Avishay Tal
2016 Computational Complexity  
We prove that random n-by-n Toeplitz (alternatively Hankel) matrices over F 2 have rigidity Ω( n 3 r 2 log n ) for rank r ≥ √ n, with high probability.  ...  This improves, for r = o(n/ log n log log n), over the Ω( n 2 r · log( n r )) bound that is known for many explicit matrices.  ...  Theorem 1.3 (on the structured rigidity of random Toeplitz/Hankel matrices). Let A ∈ F n×n 2 be a random Toeplitz/Hankel matrix.  ... 
doi:10.1007/s00037-016-0144-9 fatcat:jwuajua4nnajlojbcchrucuzhe

Acoustic multiple scattering using recursive algorithms

Feruza A. Amirkulova, Andrew N. Norris
2015 Journal of Computational Physics  
A fast recursive technique is described which takes advantage of the multilevel Block Toeplitz structure of the linear system.  ...  Numerical comparisons of CPU time and total elapsed time taken to solve the linear system using the direct LAPACK and TOEPLITZ libraries on Intel FORTRAN, show the advantage of the TOEPLITZ solver.  ...  Yuriy Gulak for helpful discussions and for providing information on available solvers including the TOEPLITZ package.  ... 
doi:10.1016/j.jcp.2015.07.031 fatcat:crusjjqsh5h5pnby5ae72ekeou

Matrix rigidity

Bruno Codenotti
2000 Linear Algebra and its Applications  
Despite more than 20 years of research, very little is known about the rigidity of matrices.  ...  The rigidity of a matrix M is the function R M (r), which, for a given r, gives the minimum number of entries of M which one has to change in order to reduce its rank to at most r.  ...  Analyze the rigidity of a Toeplitz or a circulant matrix whose entries are different indeterminates. Note that the solution to Problem 3 could have implications on the rigidity of Fourier matrices.  ... 
doi:10.1016/s0024-3795(99)00225-6 fatcat:t7fclqerxjcgncwwlllbg2hfce

Practical compressive sensing with Toeplitz and circulant matrices

Wotao Yin, Simon Morgan, Junfeng Yang, Yin Zhang, Pascal Frossard, Houqiang Li, Feng Wu, Bernd Girod, Shipeng Li, Guo Wei
2010 Visual Communications and Image Processing 2010  
Compressive sensing encodes a signal into a relatively small number of incoherent linear measurements. In theory, the optimal incoherence is achieved by completely random measurement matrices.  ...  Computational results are presented to show that Toeplitz and circulant matrices are not only as effective as random matrices for signal encoding, but also permit much faster decoding.  ...  Two such matrices are the Toeplitz and circulant matrices, which have been shown to be almost as effective as the Gaussian random matrix for CS encoding/decoding.  ... 
doi:10.1117/12.863527 dblp:conf/vcip/YinMYZ10 fatcat:447gklc2e5hq3exh7tj35d4454

Recent Progress on Matrix Rigidity – A Survey [article]

C.Ramya
2020 arXiv   pre-print
A matrix is rigid if it is far(in terms of Hamming distance) from any matrix of low rank.  ...  Although we know rigid matrices exist, obtaining explicit constructions of rigid matrices have remained a long-standing open question.  ...  Acknowledgements I am grateful to Ramprasad Saptharishi for introducing to me the concept of matrix rigidity.  ... 
arXiv:2009.09460v1 fatcat:jq73yxwe75czhp7dbvdeymzldm

Page 2537 of Mathematical Reviews Vol. , Issue 94e [page]

1994 Mathematical Reviews  
A matrix is said to be rigid if any change in each of k rows decreases the rank by no more than k units. Generalizations of this notion are given.  ...  Aurelian Gheondea (Bucharest) 94e:15051 15A57 Brualdi, Richard A. (1-WI; Madison, WI); Chavey, Keith L. (1-WI; Madison, WI) Linear spaces of Toeplitz and nilpotent matrices. (English summary) J.  ... 

Fourier and Circulant Matrices Are Not Rigid

Zeev Dvir, Allen Liu, Michael Wagner
2019 Computational Complexity Conference  
This complements a recent result of Goldreich and Tal [5] who showed that Toeplitz matrices are nontrivially rigid (but not enough for Valiant's method).  ...  The concept of matrix rigidity was first introduced by Valiant in [12] . Roughly speaking, a matrix is rigid if its rank cannot be reduced significantly by changing a small number of entries.  ...  On the other hand, this suggests that perhaps M G (f ) is a candidate for rigidity when all irreducible representations of G are large (for instance quasi-random groups [6] ).  ... 
doi:10.4230/lipics.ccc.2019.17 dblp:conf/coco/DvirL19 fatcat:4huwoz2l4jex7h7qplpwp4famm

Bohemian Matrix Geometry

Robert Corless, George Labahn, Dan Piponi, Leili Rafiee Sevyeri
2022 Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation  
A Bohemian matrix family is a set of matrices all of whose entries are drawn from a fixed, usually discrete and hence bounded, subset of a field of characteristic zero.  ...  Originally these were integershence the name, from the acronym BOunded HEight Matrix of Integers (BOHEMI)-but other kinds of entries are also interesting.  ...  We also thank her for many discussions on Bohemian matrices and colouring algorithms.  ... 
doi:10.1145/3476446.3536177 fatcat:rxeflt6umbfcpoxemv3ixnxvee

Bohemian Matrix Geometry [article]

Robert M. Corless, George Labahn, Dan Piponi, Leili Rafiee Sevyeri
2022 arXiv   pre-print
A Bohemian matrix family is a set of matrices all of whose entries are drawn from a fixed, usually discrete and hence bounded, subset of a field of characteristic zero.  ...  Originally these were integers -- hence the name, from the acronym BOunded HEight Matrix of Integers (BOHEMI) -- but other kinds of entries are also interesting.  ...  We also thank her for many discussions on Bohemian matrices and colouring algorithms. We thank Mark Giesbrecht for hours of discussion on this topic. We thank Neil J.  ... 
arXiv:2202.07769v2 fatcat:2rhafz2eafevxenyv7h4scm2lm

Statistical Estimation of EIT Electrode Contact Impedance Using a Magic Toeplitz Matrix

E. Demidenko, A. Borsic, Yuqing Wan, R. J. Halter, A. Hartov
2011 IEEE Transactions on Biomedical Engineering  
Magic Toeplitz matrix as the Neumann-to-Dirichlet map with finite width electrodes plays the central role in our linear model, called the gapZ model.  ...  The goal of this paper is to propose a fast and reliable method of simultaneous estimation of conductivity and electrode contact impedances for a homogeneous 2-D disk.  ...  He is currently a Professor at the Section of Biostatistics and Epidemiology, Dartmouth Medical School and an Adjunct Professor at the Department of Mathematics and Thayer School of Engineering, Dartmouth  ... 
doi:10.1109/tbme.2011.2125790 pmid:21402505 pmcid:PMC3233639 fatcat:hc6dsyoh3ja5rnbry26tayyz3a

Infinite Toeplitz and Laurent matrices with localized impurities

A. Böttcher, M. Embree, V.I. Sokolov
2002 Linear Algebra and its Applications  
This paper is concerned with the change of the spectra of infinite Toeplitz and Laurent matrices under perturbations in a prescribed finite set of sites.  ...  of the corresponding Laurent matrix.  ...  The matrices T (a) and L(a) are referred to as the Toeplitz matrix and the Laurent matrix with the symbol a, respectively.  ... 
doi:10.1016/s0024-3795(01)00343-3 fatcat:hvyqqjw52nhqdizxcnii4pa5jy

Optimal singular correlation matrices estimated when the sample size is less than or equal to the number of random variables

M. Vořechovský
2012 Probabilistic Engineering Mechanics  
The optimal correlation matrices are constructed for cases when the number of simulations (experiments) N sim is less than or equal to the stochastic dimension, i.e. the number of random variables (factors  ...  Many examples of optimal correlation matrices are given, both analytically and numerically.  ...  Symmetric Toeplitz matrices form a special subclass of the class of doubly symmetric matrices.  ... 
doi:10.1016/j.probengmech.2012.06.003 fatcat:zr3d3lzeujbgnkpsypssvt7cay

Reinforcement Learning with Chromatic Networks for Compact Architecture Search [article]

Xingyou Song, Krzysztof Choromanski, Jack Parker-Holder, Yunhao Tang, Wenbo Gao, Aldo Pacchiano, Tamas Sarlos, Deepali Jain, Yuxiang Yang
2021 arXiv   pre-print
matrices, while still maintaining good reward.  ...  For several RL tasks, we manage to learn colorings translating to effective policies parameterized by as few as 17 weight parameters, providing >90 over state-of-the-art compact policies based on Toeplitz  ...  Toeplitz. A toeplitz weight matrix W ∈ R a×b has a total of a+b−1 independent parameters.  ... 
arXiv:1907.06511v4 fatcat:i4d5rlvqfzgrnkgah5hlvdwhla

Fourier and Circulant Matrices are Not Rigid [article]

Zeev Dvir, Allen Liu
2021 arXiv   pre-print
The concept of matrix rigidity was first introduced by Valiant in 1977. Roughly speaking, a matrix is rigid if its rank cannot be reduced significantly by changing a small number of entries.  ...  Complexity, 2018) who showed that Toeplitz matrices are nontrivially rigid (but not enough for Valiant's method).  ...  On the other hand, this suggests that perhaps M G (f ) is a candidate for rigidity when all irreducible representations of G are large (for instance quasi-random groups [Gow08] ).  ... 
arXiv:1902.07334v3 fatcat:dxmwlnwt25e6biq4yusexkovhq
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