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Matrix rigidity of random toeplitz matrices

2016
*
Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2016
*

We prove that

doi:10.1145/2897518.2897633
dblp:conf/stoc/GoldreichT16
fatcat:xhkbzqr7mjf6be6embtsvnqd5e
*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*. ...##
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Matrix rigidity of random Toeplitz matrices

2016
*
Computational Complexity
*

We prove that

doi:10.1007/s00037-016-0144-9
fatcat:jwuajua4nnajlojbcchrucuzhe
*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*. ...##
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Acoustic multiple scattering using recursive algorithms

2015
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Journal of Computational Physics
*

A fast recursive technique is described which takes advantage

doi:10.1016/j.jcp.2015.07.031
fatcat:crusjjqsh5h5pnby5ae72ekeou
*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. ...##
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Matrix rigidity

2000
*
Linear Algebra and its Applications
*

Despite more than 20 years

doi:10.1016/s0024-3795(99)00225-6
fatcat:t7fclqerxjcgncwwlllbg2hfce
*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*. ...##
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Practical compressive sensing with Toeplitz and circulant matrices

2010
*
Visual Communications and Image Processing 2010
*

Compressive sensing encodes a signal into a relatively small number

doi:10.1117/12.863527
dblp:conf/vcip/YinMYZ10
fatcat:447gklc2e5hq3exh7tj35d4454
*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. ...##
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Recent Progress on Matrix Rigidity – A Survey
[article]

2020
*
arXiv
*
pre-print

A

arXiv:2009.09460v1
fatcat:jq73yxwe75czhp7dbvdeymzldm
*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*. ...##
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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. ...##
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Fourier and Circulant Matrices Are Not Rigid

2019
*
Computational Complexity Conference
*

This complements a recent result

doi:10.4230/lipics.ccc.2019.17
dblp:conf/coco/DvirL19
fatcat:4huwoz2l4jex7h7qplpwp4famm
*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] ). ...##
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Bohemian Matrix Geometry

2022
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Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation
*

A Bohemian

doi:10.1145/3476446.3536177
fatcat:rxeflt6umbfcpoxemv3ixnxvee
*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. ...##
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Bohemian Matrix Geometry
[article]

2022
*
arXiv
*
pre-print

A Bohemian

arXiv:2202.07769v2
fatcat:2rhafz2eafevxenyv7h4scm2lm
*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. ...##
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Statistical Estimation of EIT Electrode Contact Impedance Using a Magic Toeplitz Matrix

2011
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IEEE Transactions on Biomedical Engineering
*

Magic

doi:10.1109/tbme.2011.2125790
pmid:21402505
pmcid:PMC3233639
fatcat:hc6dsyoh3ja5rnbry26tayyz3a
*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 ...##
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Infinite Toeplitz and Laurent matrices with localized impurities

2002
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Linear Algebra and its Applications
*

This paper is concerned with the change

doi:10.1016/s0024-3795(01)00343-3
fatcat:hvyqqjw52nhqdizxcnii4pa5jy
*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. ...##
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Optimal singular correlation matrices estimated when the sample size is less than or equal to the number of random variables

2012
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Probabilistic Engineering Mechanics
*

The optimal correlation

doi:10.1016/j.probengmech.2012.06.003
fatcat:zr3d3lzeujbgnkpsypssvt7cay
*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*. ...##
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Reinforcement Learning with Chromatic Networks for Compact Architecture Search
[article]

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. ...

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Fourier and Circulant Matrices are Not Rigid
[article]

2021
*
arXiv
*
pre-print

The concept

arXiv:1902.07334v3
fatcat:dxmwlnwt25e6biq4yusexkovhq
*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] ). ...
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