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Lie algebra conjugacy [article]

Joshua A. Grochow
2011 arXiv   pre-print
A Lie algebra is semisimple if it is a direct sum of simple Lie algebras.  ...  A matrix Lie algebra is a set L of matrices such that A, B∈ L implies AB - BA ∈ L. Two matrix Lie algebras are conjugate if there is an invertible matrix M such that L_1 = M L_2 M^-1.  ...  Introduction A matrix Lie algebra is defined as a set of n × n matrices closed under the following operations: multiplication by scalars A → αA for α ∈ C, the usual matrix addition, and a multiplication-like  ... 
arXiv:1112.2012v1 fatcat:t2bgagusofcfrp3dqjefzlhlru

Designing Strassen's algorithm [article]

Joshua A. Grochow, Cristopher Moore
2017 arXiv   pre-print
Here, we give the simplest and most transparent proof of Strassen's algorithm that we are aware of, using only a simple unitary 2-design and a few easy lines of calculation.  ...  The latter construction was arrived at by a process of elimination and appears to come out of thin air.  ...  C.M. also thanks École Normale Supérieure for providing a visiting position during which some of this work was carried out.  ... 
arXiv:1708.09398v1 fatcat:i7txv335gnbexjakfc7dl2euoa

Boundaries of VP and VNP [article]

Joshua A. Grochow, Ketan D. Mulmuley, Youming Qiao
2016 arXiv   pre-print
generic semi-invariant can be computed by a circuit of polynomial size.  ...  demonstrate that the families in Newton-VP ∖ VP based on semi-invariants of quivers would have to be non-generic by showing that, for many finite quivers (including some wild ones), any Newton degeneration of a  ...  A. G. was supported by an SFI Omidyar Fellowship, K. D. M. by the NSF grant CCF-1017760, and Y. Q. by the Australian Research Council DECRA DE150100720.  ... 
arXiv:1605.02815v1 fatcat:c3eyxyeehnaanhdfl6jiopxkxy

Weisfeiler-Leman for Group Isomorphism: Action Compatibility [article]

Joshua A. Grochow, Michael Levet
2021 arXiv   pre-print
A common theme among these is that the group-theoretic structure is mostly about the action of one group on another.  ...  In particular, we show that constant-dimensional Weisfeiler-Leman only requires a constant number of rounds to identify groups in the above classes.  ...  To the best of our knowledge, WL was first utilized for group isomorphism testing by Brooksbank, Grochow, Li, Qiao, and Wilson [BGL + 19].  ... 
arXiv:2112.11487v1 fatcat:wxx2npeh3jbetlun4k5mprjdba

Matrix multiplication via matrix groups [article]

Jonah Blasiak, Henry Cohn, Joshua A. Grochow, Kevin Pratt, Chris Umans
2022 arXiv   pre-print
This is based on a representation-theoretic argument that identifies the second-smallest dimension of an irreducible representation of a group as a key parameter that determines its viability in this framework  ...  In 2003, Cohn and Umans proposed a group-theoretic approach to bounding the exponent of matrix multiplication.  ...  A. G. was partially supported by NSF CAREER award CISE-2047756, and C. U. was supported by NSF grant CCF-1815607 and a Simons Foundation Investigator award.  ... 
arXiv:2204.03826v1 fatcat:budighsph5eitkkt7rx237hnwa

Circuit complexity, proof complexity, and polynomial identity testing [article]

Joshua A. Grochow, Toniann Pitassi
2014 arXiv   pre-print
We show that either: a) Proving super-polynomial lower bounds on AC^0[p]-Frege implies VNP does not have polynomial-size circuits of depth d - a notoriously open question for d at least 4 - thus explaining  ...  We introduce a new algebraic proof system, which has tight connections to (algebraic) circuit complexity.  ...  A. G. was supported by A. Borodin's NSERC Grant # 482671.  ... 
arXiv:1404.3820v1 fatcat:2nerc6uatvdh5pqwtrhelt6jvq

Matrix multiplication algorithms from group orbits [article]

Joshua A. Grochow, Cristopher Moore
2016 arXiv   pre-print
Our constructions also suggest further patterns that could be mined for new algorithms, including a tantalizing connection with lattices.  ...  In particular, using lattices we give the most transparent proof to date of Strassen's algorithm; the same proof works for all n, to yield a decomposition with n^3 - n + 1 terms.  ...  C.M. also thanks École Normale Supérieure for providing a visiting position during which some of this work was carried out.  ... 
arXiv:1612.01527v2 fatcat:vuq7jzpjwjae3e5q5ysqsq5dtq

Incorporating Weisfeiler-Leman into algorithms for group isomorphism [article]

Peter A. Brooksbank, Joshua A. Grochow, Yinan Li, Youming Qiao, James B. Wilson
2019 arXiv   pre-print
then apply a hypergraph variant of the k-dimensional Weisfeiler-Leman technique.  ...  Recursively-refineable filters -- a generalization of subgroup series -- form the skeleton of this framework, and we refine our filter by building a hypergraph encoding low-genus quotients, to which we  ...  A. B. and J. B. W. also acknowledge the Hausdorff Institute for Mathematics, and the University of Auckland where some of this research was conducted. P. A. B., J. A. G., J. B. W., and Y.  ... 
arXiv:1905.02518v1 fatcat:okgpqyjkefbdzctsf2p5s66mhq

A quantitative definition of organismality and its application to lichen [article]

Eric Libby and Joshua Grochow and Simon DeDeo and David Wolpert
2016 arXiv   pre-print
In this approach organisms are a coarse-graining of a fine-grained dynamical model of a biological system.  ...  Here we introduce a candidate definition.  ...  Thus, the matrix a describes how to combine components of x into new macrostate individuals y: y i = a i,1 A 1 + a i,2 A 2 + a i,3 F 1 + a i,4 F 2 free-living algae and fungi (1) + a i,5 l 1,1 + a i,  ... 
arXiv:1612.00036v1 fatcat:w766mq3ol5hatjoqbo6tw33a6u

Matrix Isomorphism of Matrix Lie Algebras

Joshua A. Grochow
2012 2012 IEEE 27th Conference on Computational Complexity  
A matrix Lie algebra is a set L of matrices that is closed under linear combinations and the operation [A,B] = AB -BA.  ...  On the other hand, we give polynomial-time algorithms for other cases of MATISOLIE, which allow us to mostly derandomize a recent result of Kayal on affine equivalence of polynomials.  ...  For each pair i = j, compute the matrix [A i , A j ] = A i A j − A j A i and write it as a linear combination of the A k .  ... 
doi:10.1109/ccc.2012.34 dblp:conf/coco/Grochow12 fatcat:xhu5357knfgzloh6kqfetfrbdy

Minimum Circuit Size, Graph Isomorphism, and Related Problems [article]

Eric Allender, Joshua A. Grochow, Dieter van Melkebeek, Cristopher Moore, Andrew Morgan
2017 arXiv   pre-print
We study the computational power of deciding whether a given truth-table can be described by a circuit of a given size (the Minimum Circuit Size Problem, or MCSP for short), and of the variant denoted  ...  It yields a randomized reduction with zero-sided error from GI to MKTP.  ...  A sampler within a factor 1+δ for a distribution p on a set T is a random variable R : {0, 1} ℓ → T that induces a distribution that approximates p within a factor 1 + δ.  ... 
arXiv:1710.09806v1 fatcat:b3izhcdzznh65g7zkjuizhi6ki

On the records [article]

Andrew Berdahl, Uttam Bhat, Vanessa Ferdinand, Joshua Garland, Keyan Ghazi-Zahedi, Justin Grana, Joshua A. Grochow, Elizabeth Hobson, Yoav Kallus, Christopher P. Kempes, Artemy Kolchinsky, Daniel B. Larremore, Eric Libby, Eleanor A. Power (+1 others)
2017 arXiv   pre-print
Towards this end, we conduct a detailed analysis of a particular record-setting event: elite marathon running.  ...  Extremal records summarize the limits of the space explored by a process, and the historical progression of a record sheds light on the underlying dynamics of the process.  ...  Introduction How deep can a person dive? How tall can a building be? To answer these questions, we might look at how deep a person has ever dived, or how tall a building has ever been.  ... 
arXiv:1705.04353v2 fatcat:wwg2fdh5dzcs7eoy4njvajlgua

Which groups are amenable to proving exponent two for matrix multiplication? [article]

Jonah Blasiak, Thomas Church, Henry Cohn, Joshua A. Grochow, Chris Umans
2017 arXiv   pre-print
We prove two main results: (1) We show that a large class of nonabelian groups---nilpotent groups of bounded exponent satisfying a mild additional condition---cannot prove ω = 2 in this framework.  ...  Recently it was shown, by a generalization of the proof of the Cap Set Conjecture, that abelian groups of bounded exponent cannot prove ω = 2 in this framework, which ruled out a family of potential constructions  ...  Acknowledgments We thank Cris Moore, with help from Aaron Clauset, for the suggestion of the case of bounded variance; this was a case of the right name for a concept suggesting a better theorem than we  ... 
arXiv:1712.02302v1 fatcat:6vpiw4fomnhebatabzxjge6ovi

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
This is analogous to the Razborov-Rudich natural proofs barrier in Boolean circuit complexity, in that we rule out a large class of lower bound techniques under a derandomization assumption.  ...  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  ...  During the course of this work, J.A.G. was supported by A.  ... 
arXiv:1701.01717v1 fatcat:tfx54wjxcrdlpkm4kdkou3qthm

Unifying Known Lower Bounds via Geometric Complexity Theory

Joshua A. Grochow
2014 2014 IEEE 29th Conference on Computational Complexity (CCC)  
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.  ...  For example, the representation-theoretic viewpoint of GCT 394 Grochow cc 24 (2015) naturally provides new properties to consider in the search for new lower bounds.  ...  In the case of input matrices, test polynomials are then polynomials whose variables are the coordinates a ij of 406 Grochow cc 24 (2015) the input matrices.  ... 
doi:10.1109/ccc.2014.35 dblp:conf/coco/Grochow14 fatcat:z5eytgp64jh2pclri4tlmky53e
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