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Generating Matrix Identities and Proof Complexity
[article]

2014
*
arXiv
*
pre-print

Motivated by the fundamental

arXiv:1312.6242v4
fatcat:eb2wleirnzc2zdhqbvgva4bgpq
*lower**bounds*questions in*proof**complexity*, we initiate the study of*matrix**identities*as hard instances for strong*proof*systems. ... We present two conjectures, one about non-commutative arithmetic circuit*complexity**and*the other about*proof**complexity*, under which up to exponential-size*lower**bounds*on arithmetic*proofs*(in terms ... We are also greatly indebted to Vesselin Drensky for his help with the bibliography*and*providing us with his monograph. ...##
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Witnessing matrix identities and proof complexity

2018
*
International journal of algebra and computation
*

Finally, we present several concrete open problems about non-commutative algebraic circuits

doi:10.1142/s021819671850011x
fatcat:aevrlmha6bd5djqugsialn55fu
*and*speed-ups in*proof**complexity*, whose solution would establish stronger size*lower**bounds*on PI*proofs*of ...*matrix**identities*,*and*beyond. ... Arvind, Albert Atserias, Michael Forbes, Emil Jeřabek*and*Amir Shpilka for useful discussions related to this work. ...##
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General Capacity Bounds for Spatially Correlated Rician MIMO Channels

2005
*
IEEE Transactions on Information Theory
*

We consider the

doi:10.1109/tit.2005.853325
fatcat:z6oofa2jljakzl2dnubk6yg6v4
*general*case with double-sided correlation*and*arbitrary rank channel means. We derive tight upper*and**lower**bounds*on the ergodic capacity. ... The*bounds*are shown to reduce to known results in cases of independent*and**identically*distributed (i.i.d.)*and*correlated Rayleigh MIMO channels. ... The superscripts ,*and*indicate*matrix*transpose,*complex*conjugate,*and**complex*conjugate transpose respectively. The matrices*and*denote a*identity**and*a all-zero*matrix*, respectively. ...##
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A Lower Bound on Determinantal Complexity
[article]

2021
*
arXiv
*
pre-print

For every n-variate polynomial of degree d, the determinantal

arXiv:2009.02452v2
fatcat:nct22zjlgzdpzpjp6ahgs2xixi
*complexity*is trivially at least d,*and*it is a long standing open problem to prove a*lower**bound*which is super linear in max{n,d}. ... Our result is the first*lower**bound*for any explicit polynomial which is bigger by a constant factor than max{n,d},*and*improves upon the prior best*bound*of n + 1, proved by Alper, Bogart*and*Velasco ... Acknowledgment Mrinal thanks Ramprasad Saptharishi for various discussions on determinantal*complexity*over the years,*and*in particular for explaining the*proof*of the result of Mignon*and*Ressayre to ...##
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Vector-Matrix-Vector Queries for Solving Linear Algebra, Statistics, and Graph Problems

2020
*
International Workshop on Approximation Algorithms for Combinatorial Optimization
*

Our work is exploratory

doi:10.4230/lipics.approx/random.2020.26
dblp:conf/approx/RashtchianWZ20
fatcat:lnwrpizapnhwxicklk5kbrailu
*and*broad,*and*we provide new upper*and**lower**bounds*for a wide variety of problems, spanning linear algebra, statistics,*and*graphs. ... We consider the*general*problem of learning about a*matrix*through vector-*matrix*-vector queries. ... Then to determine whether M is a unitary*matrix*in deterministic case, the*lower**bound*of query*complexity*is Ω(n 2 / log n).*Proof*. We reduce the problem to Disjointness. ...##
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Algebraic Methods in Computational Complexity (Dagstuhl Seminar 16411)

2017
*
Dagstuhl Reports
*

Indeed, many of the deepest

doi:10.4230/dagrep.6.10.13
dblp:journals/dagstuhl-reports/KabanetsTTU16
fatcat:mxexvi3xmngw7iwfc2bodtsd3y
*and*most powerful results in Computational*Complexity*rely on algebraic*proof*techniques. ... would lead to a*general*result. ... We prove a*general*result*lower**bounding*the randomized communication*complexity*of the elimination problem for f using its discrepancy. ...##
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On Matrix Multiplication and Polynomial Identity Testing
[article]

2022
*
arXiv
*
pre-print

We show that

arXiv:2208.01078v1
fatcat:676drnxzmvduhnk5takbdztiqm
*lower**bounds*on the border rank of*matrix*multiplication can be used to non-trivially derandomize polynomial*identity*testing for small algebraic circuits. ... If the*matrix*multiplication exponent ω is not 2, our*generator*has seed length O(n^1 - ε)*and*hits circuits of size O(n^1 + δ) for sufficiently small ε, δ > 0. ... Acknowledgments We thank Shubhang Kulkarni for telling us about the work of Barahona*and*Pulleyblank [BP87] . ...##
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Quantum Query Complexity of Multilinear Identity Testing

2009
*
Symposium on Theoretical Aspects of Computer Science
*

Towards a

doi:10.4230/lipics.stacs.2009.1801
dblp:conf/stacs/ArvindM09
fatcat:nyqom4g4rrch7ltsy76getof44
*lower**bound*, we also show a reduction from a version of the collision problem (which is well studied in quantum computation) to a variant of this problem. ... Motivated by the quantum algorithm for testing commutativity of black-box groups , we study the following problem: Given a black-box finite ring by an additive*generating*set*and*a multilinear polynomial ... Acknowledgement We thank Ashwin Nayak for comments*and*suggestions. We are grateful to the anonymous STACS'09 referees for useful comments. ...##
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Quantum query complexity with matrix-vector products
[article]

2021
*
arXiv
*
pre-print

We demonstrate this by showing equivalence between models that provide

arXiv:2102.11349v2
fatcat:wftqhtmzbvf2rfvvpcjlakkphm
*matrix*-vector products, vector-*matrix*products,*and*vector-*matrix*-vector products, whereas the power of these models can vary significantly ... On the other hand, we show that for some problems, such as computing the parities of rows or columns or deciding if there are two*identical*rows or columns, quantum computers provide exponential speedup ... Acknowledgments We thank Robin Kothari for bringing our attention to work on classical algorithms in the matrixvector*and*vector-*matrix*-vector query models,*and*for providing feedback on an initial version ...##
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Algorithms for Pattern Containment in 0-1 Matrices
[article]

2017
*
arXiv
*
pre-print

In this paper, we present optimal algorithms to determine when an n × n

arXiv:1704.05207v1
fatcat:mszdewacprf3hateghzpnwpnyu
*matrix*A contains a given pattern P when P is a column of all ones, an*identity**matrix*, a tuple*identity**matrix*, an L-shaped pattern ... We say a zero-one*matrix*A avoids another zero-one*matrix*P if no submatrix of A can be transformed to P by changing some ones to zeros. ... However, our algorithm for*general*cross patterns is more*complex*than our algorithm above for L-shaped patterns.*Proof*. ...##
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On the Symmetries of and Equivalence Test for Design Polynomials

2019
*
International Symposium on Mathematical Foundations of Computer Science
*

The family of polynomials N W := {NW d,k : d is a prime}

doi:10.4230/lipics.mfcs.2019.53
dblp:conf/mfcs/GuptaS19
fatcat:46awkhafpfghhnr5p47z6kc4gq
*and*close variants of it have been used as hard explicit polynomial families in several recent arithmetic circuit*lower**bound**proofs*. ... Characterization of polynomials by their symmetries plays a central role in the geometric*complexity*theory program. Here, we answer the first two questions*and*partially answer the third. ... NG would also like to thank Anuj Tawari for his time in sitting through a few presentations on the*proof*of Theorem 4. ...##
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Lower Bounds for Dynamic Algebraic Problems
[chapter]

1999
*
Lecture Notes in Computer Science
*

We also show linear

doi:10.1007/3-540-49116-3_34
fatcat:6vfarspqkfa2jci345ue4q5biu
*lower**bounds*for dynamic determinant,*matrix*adjoint*and**matrix*inverse*and*an Ω( √ n)*lower**bound*for the elementary symmetric functions. ... Using this technique, we show optimal Ω(n)*bounds*for dynamic*matrix*-vector product, dynamic*matrix*multiplication*and*dynamic discriminant*and*an Ω( √ n)*lower**bound*for dynamic polynomial multiplication ... Thus, the Ω(n)*lower**bound*also holds for*matrix*adjoint*and**matrix*inverse. ...##
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Vector-Matrix-Vector Queries for Solving Linear Algebra, Statistics, and Graph Problems
[article]

2020
*
arXiv
*
pre-print

Our work is exploratory

arXiv:2006.14015v1
fatcat:ihl3przjlre4jpeus2w6mr4eza
*and*broad,*and*we provide new upper*and**lower**bounds*for a wide variety of problems, spanning linear algebra, statistics,*and*graphs. ... Many of our results are nearly tight,*and*we use diverse techniques from linear algebra, randomized algorithms,*and*communication*complexity*. ... Then to determine whether M is a unitary*matrix*with a constant probability, the*lower**bound*of query*complexity*is Ω(n/ log n)*and*the upper*bound*is O(n).*Proof*. ...##
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Proving Lower Bounds Via Pseudo-random Generators
[chapter]

2005
*
Lecture Notes in Computer Science
*

In this paper, we formalize two stepwise approaches, based on pseudo-random

doi:10.1007/11590156_6
fatcat:utnfe6pyvfczxkyaqpy6gchrdu
*generators*, for proving P = NP*and*its arithmetic analog: Permanent requires superpolynomial sized arithmetic circuits. ... Introduction The central aim of*complexity*theory is to prove*lower**bounds*on the*complexity*of problems. ... Even single variable circuits can compute very*complex*polynomials. Kabanets*and*Impagliazzo [11] showed a connection between polynomial*identity*testing*and**lower**bounds*on arithmetic circuits. ...##
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Towards an algebraic natural proofs barrier via polynomial identity testing
[article]

2017
*
arXiv
*
pre-print

We observe that a certain kind of algebraic

arXiv:1701.01717v1
fatcat:tfx54wjxcrdlpkm4kdkou3qthm
*proof*- which covers essentially all known algebraic circuit*lower**bounds*to date - cannot be used to prove*lower**bounds*against VP if*and*only if what we call ... We also discuss connections between this algebraic natural*proofs*barrier, geometric*complexity*theory,*and*(algebraic)*proof**complexity*. ... Borodin's NSERC Grant # 482671, an Omidyar Fellowship from the Santa Fe Institute,*and*NSF grant DMS-1620484; M.S. was supported by NSF grant CCF-1218711*and*by Simons Foundation Award 332622;*and*S.S. ...
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