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Lie algebra conjugacy
[article]

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*→ α

*A*for α ∈ C, the usual matrix addition, and

*a*multiplication-like ...

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Designing Strassen's algorithm
[article]

2017
*
arXiv
*
pre-print

Here, we give the simplest and most transparent proof of Strassen's algorithm that we are aware of, using only

arXiv:1708.09398v1
fatcat:i7txv335gnbexjakfc7dl2euoa
*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. ...##
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Boundaries of VP and VNP
[article]

2016
*
arXiv
*
pre-print

generic semi-invariant can be computed by

arXiv:1605.02815v1
fatcat:c3eyxyeehnaanhdfl6jiopxkxy
*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. ...##
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Weisfeiler-Leman for Group Isomorphism: Action Compatibility
[article]

2021
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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]. ...

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Matrix multiplication via matrix groups
[article]

2022
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arXiv
*
pre-print

This is based on

arXiv:2204.03826v1
fatcat:budighsph5eitkkt7rx237hnwa
*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. ...##
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Circuit complexity, proof complexity, and polynomial identity testing
[article]

2014
*
arXiv
*
pre-print

We show that either:

arXiv:1404.3820v1
fatcat:2nerc6uatvdh5pqwtrhelt6jvq
*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. ...##
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Matrix multiplication algorithms from group orbits
[article]

2016
*
arXiv
*
pre-print

Our constructions also suggest further patterns that could be mined for new algorithms, including

arXiv:1612.01527v2
fatcat:vuq7jzpjwjae3e5q5ysqsq5dtq
*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. ...##
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Incorporating Weisfeiler-Leman into algorithms for group isomorphism
[article]

2019
*
arXiv
*
pre-print

then apply

arXiv:1905.02518v1
fatcat:okgpqyjkefbdzctsf2p5s66mhq
*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. ...##
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A quantitative definition of organismality and its application to lichen
[article]

2016
*
arXiv
*
pre-print

In this approach organisms are

arXiv:1612.00036v1
fatcat:w766mq3ol5hatjoqbo6tw33a6u
*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, ...##
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Matrix Isomorphism of Matrix Lie Algebras

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 . ...

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

2017
*
arXiv
*
pre-print

We study the computational power of deciding whether

arXiv:1710.09806v1
fatcat:b3izhcdzznh65g7zkjuizhi6ki
*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. ...##
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On the records
[article]

2017
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arXiv
*
pre-print

Towards this end, we conduct

arXiv:1705.04353v2
fatcat:wwg2fdh5dzcs7eoy4njvajlgua
*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. ...##
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Which groups are amenable to proving exponent two for matrix multiplication?
[article]

2017
*
arXiv
*
pre-print

We prove two main results: (1) We show that

arXiv:1712.02302v1
fatcat:6vpiw4fomnhebatabzxjge6ovi
*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 ...##
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Towards an algebraic natural proofs barrier via polynomial identity testing
[article]

2017
*
arXiv
*
pre-print

This is analogous to the Razborov-Rudich natural proofs barrier in Boolean circuit complexity, in that we rule out

arXiv:1701.01717v1
fatcat:tfx54wjxcrdlpkm4kdkou3qthm
*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*. ...##
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Unifying Known Lower Bounds via Geometric Complexity Theory

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

This enables us to expose

doi:10.1109/ccc.2014.35
dblp:conf/coco/Grochow14
fatcat:z5eytgp64jh2pclri4tlmky53e
*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. ...
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