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A Polynomial Degree Bound on Equations for Non-Rigid Matrices and Small Linear Circuits

2022
*
ACM Transactions on Computation Theory
*

Our methods are elementary

doi:10.1145/3543685
fatcat:4fdzhmpdyzag5cqhnqcsnyra4y
*and*short*and*rely*on**a**polynomial*map of Shpilka*and*Volkovich [21] to construct low*degree*"universal" maps*for**non*-*rigid**matrices**and**small**linear**circuits*. ... We show that there is an*equation*of*degree*at most poly( n )*for*the (Zariski closure of the) set of the*non*-*rigid**matrices*: that is, we show that*for*every large enough field \(\mathbb {F} \) , there ... ACKNOWLEDGMENTS We thank the anonymous reviewers*for*their careful reading*and*thoughtful comments*on**a*preliminary version of this paper. ...##
###
A Polynomial Degree Bound on Equations for Non-Rigid Matrices and Small Linear Circuits

2021

Our methods are elementary

doi:10.4230/lipics.itcs.2021.9
fatcat:tulgfxpdfvf6hlmkyjrt32sdni
*and*short*and*rely*on**a**polynomial*map of Shpilka*and*Volkovich [Amir Shpilka*and*Ilya Volkovich, 2015] to construct low*degree*"universal" maps*for**non*-*rigid**matrices**and**small*... We show that there is an*equation*of*degree*at most poly(n)*for*the (Zariski closure of the) set of the*non*-*rigid**matrices*: that is, we show that*for*every large enough field 𝔽, there is*a**non*-zero n² ... I T C S 2 0 2 1 9:8*Polynomial**Degree**Bound**on**Equations**for**Non*-*Rigid**Matrices*Proof of Corollary 3. ...##
###
A Polynomial Degree Bound on Equations of Non-rigid Matrices and Small Linear Circuits
[article]

2020
*
arXiv
*
pre-print

Our methods are elementary

arXiv:2003.12938v2
fatcat:bmordmypzvbvlcnk332sd4z6ly
*and*short*and*rely*on**a**polynomial*map of Shpilka*and*Volkovich [SV15] to construct low*degree*"universal" maps*for**non*-*rigid**matrices**and**small**linear**circuits*. ... We show that there is*a*defining*equation*of*degree*at most 𝗉𝗈𝗅𝗒(n)*for*the (Zariski closure of the) set of the*non*-*rigid**matrices*: that is, we show that*for*every large enough field 𝔽, there is*a*... This argument provides*a*(different)*equation*of*polynomial**degree**for*each irreducible component of the variety of*non*-*rigid**matrices*. ♦*Degree*Upper*bound**for**Matrices*with*a**Small**Circuit*In this ...##
###
Lower Bounds for Matrix Factorization

2020
*
Computational Complexity Conference
*

*circuits*which compute

*linear*transformations, matrix

*rigidity*

*and*data structure lower

*bounds*. ... This improves upon the prior best lower

*bounds*

*for*this problem, which are barely super-

*linear*,

*and*were obtained by

*a*long line of research based

*on*the study of super-concentrators (albeit at the cost ...

*A*lower

*bound*of s

*on*the size of depth d

*linear*

*circuits*computing the

*linear*transformation Ax implies

*a*lower

*bound*of Ω(s)

*for*depth Ω(d) algebraic

*circuits*computing the

*degree*-2

*polynomial*y T Ax ...

##
###
Lower Bounds for Matrix Factorization
[article]

2019
*
arXiv
*
pre-print

*circuits*which compute

*linear*transformations, matrix

*rigidity*

*and*data structure lower

*bounds*. ... This improves upon the prior best lower

*bounds*

*for*this problem, which are barely super-

*linear*,

*and*were obtained by

*a*long line of research based

*on*the study of super-concentrators (albeit at the cost ... We also thank Rohit Gurjar, Nutan Limaye, Srikanth Srinivasan

*and*Joel Tropp

*for*helpful discussions. ...

##
###
Probabilistic Rank and Matrix Rigidity
[article]

2017
*
arXiv
*
pre-print

We also show

arXiv:1611.05558v2
fatcat:lqij2wpjzrduxoa3g5iswj6q2a
*non*-trivial*rigidity*upper*bounds**for*H_n with smaller target rank. Matrix*Rigidity**and*Threshold*Circuit*Lower*Bounds*. ... We give surprising upper*bounds**on*the*rigidity*of*a*family of*matrices*whose*rigidity*has been extensively studied,*and*was conjectured to be highly*rigid*. ... s conjectures*and*results*on**non*-*rigidity*at Banff (BIRS) in August 2016. ...##
###
Recent Progress on Matrix Rigidity – A Survey
[article]

2020
*
arXiv
*
pre-print

In this survey, we present

arXiv:2009.09460v1
fatcat:jq73yxwe75czhp7dbvdeymzldm
*a*selected set of results that highlight recent progress*on*matrix*rigidity**and*its remarkable connections to other areas in theoretical computer science. ... Although we know*rigid**matrices*exist, obtaining explicit constructions of*rigid**matrices*have remained*a*long-standing open question. ... I thank Ramprasad Saptharishi, Anamay Tengse*and*Prerona Chatterjee*for*numerous technical discussions*on*the various papers presented in this article. ...##
###
Matrix rigidity of random toeplitz matrices

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

This improves,

doi:10.1145/2897518.2897633
dblp:conf/stoc/GoldreichT16
fatcat:xhkbzqr7mjf6be6embtsvnqd5e
*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*. ... 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. ... We will present two tests:*One**for*AN-complexity failing most*matrices*taken from*a**small*-biased space,*and**one**for*AN2-complexity failing most Toeplitz*matrices*. ...##
###
Matrix rigidity of random Toeplitz matrices

2016
*
Computational Complexity
*

This improves,

doi:10.1007/s00037-016-0144-9
fatcat:jwuajua4nnajlojbcchrucuzhe
*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*. ... 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. ... We will present two tests:*One**for*AN-complexity failing most*matrices*taken from*a**small*-biased space,*and**one**for*AN2-complexity failing most Toeplitz*matrices*. ...##
###
Complexity of Linear Circuits and Geometry

2015
*
Foundations of Computational Mathematics
*

of

doi:10.1007/s10208-015-9258-8
fatcat:yfbantsj2bfzfnyzilreo3x2cm
*matrices*that are expected to have super-*linear**rigidity*,*and*(4) prove results about the ideals*and**degrees*of cones that are of interest in their own right. ... In particular, we (1) exhibit many*non*-obvious*equations*testing*for*(border)*rigidity*, (2) compute*degrees*of varieties associated with*rigidity*, (3) describe algebraic varieties associated with families ... Acknowledgments We thank the anonymous referees*for*very careful reading*and*numerous useful suggestions. ...##
###
Complexity of linear circuits and geometry
[article]

2015
*
arXiv
*
pre-print

We (i) exhibit many

arXiv:1310.1362v2
fatcat:oxe5yy6xcnbzdngmda3petvlfu
*non*-obvious*equations*testing*for*(border)*rigidity*, (ii) compute*degrees*of varieties associated to*rigidity*, (iii) describe algebraic varieties associated to families of*matrices*... that are expected to have super-*linear**rigidity*,*and*(iv) prove results about the ideals*and**degrees*of cones that are of interest in their own right. ... Given*a**polynomial*P*on*the space of n × n*matrices*that vanishes*on*all*matrices*of low*rigidity*(complexity),*and**a*matrix*A*such that P (*A*) = 0,*one*obtains*a*lower*bound**on*the*rigidity*(complexity ...##
###
Matrix Rigidity from the Viewpoint of Parameterized Complexity

2018
*
SIAM Journal on Discrete Mathematics
*

*Rigidity*is

*a*classical concept in Computational Complexity Theory: constructions of

*rigid*

*matrices*are known to imply lower

*bounds*of significant importance relating to arithmetic

*circuits*. ... The steps of procedure Column-Reduction are all computable in

*polynomial*time,

*and*therefore Matrix-Reduction runs in

*polynomial*time. We now prove the desired properties

*one*by

*one*. ... Valiant [22] presented the notion of the

*rigidity*of

*a*matrix as

*a*means to prove lower

*bounds*

*for*

*linear*algebraic

*circuits*. ...

##
###
On the Existence of Algebraically Natural Proofs
[article]

2021
*
arXiv
*
pre-print

Our proofs are elementary

arXiv:2004.14147v2
fatcat:4ikspq4ebjeuxnscucjccfx4by
*and*rely*on*the existence of (*non*-explicit) hitting sets*for*VP (*and*VNP) to show that there are efficiently constructible, low*degree**equations**for*these classes. ...*For*every constant c > 0, we show that there is*a*family P_N, c of*polynomials*whose*degree**and*algebraic*circuit*complexity are*polynomially**bounded*in the number of variables, that satisfies the following ... We also thank Ben Lee Volk*for**a*careful reading of the paper*and*insightful comments which helped improve the presentation. ...##
###
Fast, Algebraic Multivariate Multipoint Evaluation in Small Characteristic and Applications
[article]

2022
*
arXiv
*
pre-print

*and*this algebraic structure naturally leads to the applications to data structure upper

*bounds*

*for*

*polynomial*evaluation

*and*to an upper

*bound*

*on*the

*rigidity*of Vandermonde

*matrices*. ... In

*a*significant improvement to the state of art

*for*this problem, Umans

*and*Kedlaya & Umans gave nearly

*linear*time algorithms

*for*this problem over field of

*small*characteristic

*and*over all finite fields ... We also thank Ben Lund

*for*helpful discussions

*and*references

*on*the finite fields Kakeya problem

*and*

*for*pointing us to the relevant literature

*on*Furstenberg sets, both of which indirectly played

*a*role ...

##
###
Unifying Known Lower Bounds via Geometric Complexity Theory

2014
*
2014 IEEE 29th Conference on Computational Complexity (CCC)
*

partial derivatives technique (Nisan-Wigderson), the results of Razborov

doi:10.1109/ccc.2014.35
dblp:conf/coco/Grochow14
fatcat:z5eytgp64jh2pclri4tlmky53e
*and*Smolensky*on*AC 0 [p], multilinear formula*and**circuit*size lower*bounds*(Raz et al.), the*degree**bound*(Strassen, Baur-Strassen ... This enables us to expose*a*new viewpoint*on*GCT, whereby it is*a*natural unification of known results*and*broad generalization of known techniques. ... Landsberg, Ketan Mulmuley, Toni Pitassi, Peter Scheiblechner, Chris Umans, Alasdair Urquhart, Ryan Williams,*and*Yiwei She*for*useful discussions. In particular, Williams suggested the ...
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