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Generating Matrix Identities and Proof Complexity [article]

Fu Li, Iddo Tzameret
2014 arXiv   pre-print
Motivated by the fundamental 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.  ... 
arXiv:1312.6242v4 fatcat:eb2wleirnzc2zdhqbvgva4bgpq

Witnessing matrix identities and proof complexity

Fu Li, Iddo Tzameret
2018 International journal of algebra and computation  
Finally, we present several concrete open problems about non-commutative algebraic circuits 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.  ... 
doi:10.1142/s021819671850011x fatcat:aevrlmha6bd5djqugsialn55fu

General Capacity Bounds for Spatially Correlated Rician MIMO Channels

M.R. McKay, I.B. Collings
2005 IEEE Transactions on Information Theory  
We consider the 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.  ... 
doi:10.1109/tit.2005.853325 fatcat:z6oofa2jljakzl2dnubk6yg6v4

A Lower Bound on Determinantal Complexity [article]

Mrinal Kumar, Ben Lee Volk
2021 arXiv   pre-print
For every n-variate polynomial of degree d, the determinantal 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  ... 
arXiv:2009.02452v2 fatcat:nct22zjlgzdpzpjp6ahgs2xixi

Vector-Matrix-Vector Queries for Solving Linear Algebra, Statistics, and Graph Problems

Cyrus Rashtchian, David P. Woodruff, Hanlin Zhu, Raghu Meka, Jarosław Byrka
2020 International Workshop on Approximation Algorithms for Combinatorial Optimization  
Our work is exploratory 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.  ... 
doi:10.4230/lipics.approx/random.2020.26 dblp:conf/approx/RashtchianWZ20 fatcat:lnwrpizapnhwxicklk5kbrailu

Algebraic Methods in Computational Complexity (Dagstuhl Seminar 16411)

Valentine Kabanets, Thomas Thierauf, Jacobo Tóran, Christopher Umans, Marc Herbstritt
2017 Dagstuhl Reports  
Indeed, many of the deepest 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.  ... 
doi:10.4230/dagrep.6.10.13 dblp:journals/dagstuhl-reports/KabanetsTTU16 fatcat:mxexvi3xmngw7iwfc2bodtsd3y

On Matrix Multiplication and Polynomial Identity Testing [article]

Robert Andrews
2022 arXiv   pre-print
We show that 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] .  ... 
arXiv:2208.01078v1 fatcat:676drnxzmvduhnk5takbdztiqm

Quantum Query Complexity of Multilinear Identity Testing

Vikraman Arvind, Partha Mukhopadhyay, Marc Herbstritt
2009 Symposium on Theoretical Aspects of Computer Science  
Towards a 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.  ... 
doi:10.4230/lipics.stacs.2009.1801 dblp:conf/stacs/ArvindM09 fatcat:nyqom4g4rrch7ltsy76getof44

Quantum query complexity with matrix-vector products [article]

Andrew M. Childs, Shih-Han Hung, Tongyang Li
2021 arXiv   pre-print
We demonstrate this by showing equivalence between models that provide 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  ... 
arXiv:2102.11349v2 fatcat:wftqhtmzbvf2rfvvpcjlakkphm

Algorithms for Pattern Containment in 0-1 Matrices [article]

P.A. CrowdMath
2017 arXiv   pre-print
In this paper, we present optimal algorithms to determine when an n × n 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.  ... 
arXiv:1704.05207v1 fatcat:mszdewacprf3hateghzpnwpnyu

On the Symmetries of and Equivalence Test for Design Polynomials

Nikhil Gupta, Chandan Saha, Michael Wagner
2019 International Symposium on Mathematical Foundations of Computer Science  
The family of polynomials N W := {NW d,k : d is a prime} 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.  ... 
doi:10.4230/lipics.mfcs.2019.53 dblp:conf/mfcs/GuptaS19 fatcat:46awkhafpfghhnr5p47z6kc4gq

Lower Bounds for Dynamic Algebraic Problems [chapter]

Gudmund Skovbjerg Frandsen, Johan P. Hansen, Peter Bro Miltersen
1999 Lecture Notes in Computer Science  
We also show linear 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.  ... 
doi:10.1007/3-540-49116-3_34 fatcat:6vfarspqkfa2jci345ue4q5biu

Vector-Matrix-Vector Queries for Solving Linear Algebra, Statistics, and Graph Problems [article]

Cyrus Rashtchian, David P. Woodruff, Hanlin Zhu
2020 arXiv   pre-print
Our work is exploratory 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.  ... 
arXiv:2006.14015v1 fatcat:ihl3przjlre4jpeus2w6mr4eza

Proving Lower Bounds Via Pseudo-random Generators [chapter]

Manindra Agrawal
2005 Lecture Notes in Computer Science  
In this paper, we formalize two stepwise approaches, based on pseudo-random 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.  ... 
doi:10.1007/11590156_6 fatcat:utnfe6pyvfczxkyaqpy6gchrdu

Towards an algebraic natural proofs barrier via polynomial identity testing [article]

Joshua A. Grochow and Mrinal Kumar and Michael Saks and Shubhangi Saraf
2017 arXiv   pre-print
We observe that a certain kind of algebraic 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.  ... 
arXiv:1701.01717v1 fatcat:tfx54wjxcrdlpkm4kdkou3qthm
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