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Small Space Analogues of Valiant's Classes and the Limitations of Skew Formulas

2011
*
Computational Complexity
*

We also show that exponential sums

doi:10.1007/s00037-011-0024-2
fatcat:bnro4engmva25nobjrwr2cxvmq
*of*a*skew**formula*cannot represent*the*determinant polynomial. ... We introduce*the**class*VL as an algebraic variant*of*deterministic log-*space*L; VL is a subclass*of*VP. ... Acknowledgments*The*authors are indebted to*the*anonymous reviewers for very detailed comments, which helped to significantly improve*the*presentation*of**the*results,*and*for pointing out a flaw in an ...##
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An Overview of Mathematical Issues Arising in the Geometric Complexity Theory Approach to $\mathbf{VP}\neq\mathbf{VNP}$

2011
*
SIAM journal on computing (Print)
*

We discuss

doi:10.1137/090765328
fatcat:mf7w367eqreopbr76y5v6ihtaq
*the*geometry*of*orbit closures*and**the*asymptotic behavior*of*Kronecker coefficients in*the*context*of**the*geometric complexity theory program to prove a variant*of**Valiant's*algebraic*analogue*... We also describe*the*precise separation*of*complexity*classes*that their program proposes to demonstrate. ... This paper is an outgrowth*of**the*AIM workshop Geometry*and*representation theory*of*tensors for computer science, statistics*and*other areas, July 21-25, 2008,*and**the*authors gratefully thank AIM*and*...##
###
P versus NP and geometry
[article]

2010
*
arXiv
*
pre-print

NP, present several results that illustrate

arXiv:0910.2443v2
fatcat:aml4xmebqjfrxonz5bcwusafpa
*the*role*of*group actions in complexity theory,*and*make a first step towards completely geometric definitions*of*complexity*classes*. ... I describe three geometric approaches to resolving variants*of*P v. ... VNP is*Valiant's*algebraic*analogue**of*NP, see §6 for a definition. ...##
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Progress in Computational Complexity Theory

2005
*
Journal of Computer Science and Technology
*

connectivity problem; deterministic polynomial-time primality test; lattice complexity, worst-case to averagecase reductions; pseudorandomness

doi:10.1007/s11390-005-0735-4
fatcat:67sjhrf4djc3hkhmyla45r63n4
*and*extractor constructions;*and**Valiant's*new theory*of*... We will discuss*the*PCP Theorem, its implications to inapproximability on combinatorial optimization problems;*space*bounded computations, especially deterministic logspace algorithm for undirected graph ... We also thank Vinay Choudhary, Rakesh Kumar*and*Anand Sinha for many interesting discussions,*and*to Vinay Choudhary for his help in*the*bibliography. ...##
###
The Computational Complexity of Plethysm Coefficients

2020
*
Computational Complexity
*

*The*role

*of*

*the*Kronecker coefficients in that setting is taken by

*the*so-called plethysm coefficients: These are

*the*multiplicities in

*the*coordinate rings

*of*

*spaces*

*of*polynomials. ... While tensors are an interesting test bed for GCT ideas,

*the*far-away goal is

*the*separation

*of*algebraic complexity

*classes*. ...

*The*author cc

*The*Complexity

*of*Plethysm Coefficients Page 37

*of*43 8 ...

##
###
Deterministic Black-Box Identity Testing $pi$-Ordered Algebraic Branching Programs

2010
*
Foundations of Software Technology and Theoretical Computer Science
*

Our next set

doi:10.4230/lipics.fsttcs.2010.296
dblp:conf/fsttcs/JansenQS10
fatcat:bfez3ibuxrcrjbzj5fdicb4dqq
*of*results investigates*the*computational*limitations**of*OABPs. It is shown that any OABP computing*the*determinant or permanent requires size Ω(2 n /n)*and*read Ω(2 n /n 2 ). ... One can think*of*OABPs as being*the*arithmetic*analogue**of*ordered binary decision diagrams (OBDDs). We say an ABP A is*of*read r, if any variable appears at most r times in A. ...*classes*VP*and*VNP are distinct. ...##
###
Linear Stochastic Systems: A White Noise Approach

2009
*
Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications
*

Kaashoek Vrije Universiteit,

doi:10.1007/s10440-009-9461-1
fatcat:2sbydn4vobgsnfq7szugarnqpy
*The*Netherlands Simple examples, due to Gohberg*and*Heinig [1] , show that for regular matrix polynomials it can happen that*the*null*space**of**the*block matrix*analogue**of*... However, if D is not a vector*space*over C then κ D (A, λ) provides only incomplete information about*the*mobility*of*λ under*small*perturbations from D. ...##
###
Circuit complexity, proof complexity, and polynomial identity testing
[article]

2014
*
arXiv
*
pre-print

than lower bounds on

arXiv:1404.3820v1
fatcat:2nerc6uatvdh5pqwtrhelt6jvq
*the*corresponding circuit*classes*. ...*the*difficulty*of*lower bounds on AC^0[p]-Frege, or b) AC^0[p]-Frege cannot efficiently prove*the*depth d PIT axioms,*and*hence we have a lower bound on AC^0[p]-Frege. ... Acknowledgments We thank David Liu for many interesting discussions,*and*for collaborating with us on some*of**the*open questions posed in this paper. ...##
###
Circuit Complexity, Proof Complexity, and Polynomial Identity Testing

2014
*
2014 IEEE 55th Annual Symposium on Foundations of Computer Science
*

than lower bounds on

doi:10.1109/focs.2014.20
dblp:conf/focs/GrochowP14
fatcat:mhmjd72cr5gmfdgtfl4zgsfhsi
*the*corresponding circuit*classes*. ... Circuit Complexity, Proof Complexity,*and*PIT -J. A. Grochow*and*T. Pitassi From*the*proof*of*this result, together with one*of*our simulation results (Proposition 2.2), we also get: Corollary 1.2. ... Acknowledgments We thank David Liu for many interesting discussions,*and*for collaborating with us on some*of**the*open questions posed in this paper. ...##
###
Geometric Complexity Theory: Introduction
[article]

2007
*
arXiv
*
pre-print

Part I consists

arXiv:0709.0746v1
fatcat:dm4qzrceqzhwvpfbzh2a4izi24
*of**the*lecture notes for*the*course given by*the*first author in*the*spring quarter, 2007. It gives introduction to*the*basic structure*of*GCT. ... These lecture notes in conjunction with*the*article GCTflip1, which describes in detail*the*basic plan*of*GCT based on*the*principle called*the*flip, should provide a high level picture*of*GCT assuming ... Let C*class*(G) be*the**space**of**class*functions on (G),*and*let (α, β) = 1 |G| g α(g)β(g) be*the*Hermitian form on C*class*Lemma 3.7. ...##
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Recent progress on scaling algorithms and applications
[article]

2018
*
arXiv
*
pre-print

This survey is intended for a general computer science audience,

arXiv:1808.09669v1
fatcat:eohzepzccfhandivheqyvslej4
*and**the*only background required is basic knowledge*of*calculus*and*linear algebra, thereby making it accessible to graduate students*and*... Scaling problems have a rich*and*diverse history,*and*thereby have found numerous applications in several fields*of*science*and*engineering. ... Acknowledgements We thank Vikraman Arvind for inviting us to write this survey for*the*EATCS complexity column (June 2018 issue)*and*Avi Wigderson for providing helpful comments on an earlier version*of*...##
###
On Agnostic Learning of Parities, Monomials, and Halfspaces

2009
*
SIAM journal on computing (Print)
*

We study

doi:10.1137/070684914
fatcat:on6brq4wyfchrl55fsdbhtevuq
*the*learnability*of*several fundamental concept*classes*in*the*agnostic learning framework*of*Haussler [Hau92]*and*Kearns et al. [KSS94]. ... We use similar techniques to reduce learning*of*two other fundamental concept*classes*under*the*uniform distribution to learning*of*noisy parities. ... We would also like to thank Shaili Jain*and**the*anonymous referees*of*CCC '06*and*FOCS '06 for numerous helpful remarks, one*of*which simplified*the*proof*of*Theorem 8. ...##
###
Interactions of Computational Complexity Theory and Mathematics
[article]

2017
*
arXiv
*
pre-print

In my view,

arXiv:1710.09780v1
fatcat:js6k4t7gefcutfus6te4vxx6ty
*the*breadth, depth, beauty*and*novelty*of*these connections is inspiring,*and*speaks to a great potential*of*future interactions (which indeed, are quickly expanding). We aim for variety. ... We give short, simple descriptions (without proofs or much technical detail)*of*ideas, motivations, results*and*connections; this will hopefully entice*the*reader to dig deeper. ... p ∈ F[X] G there is a t-variate polynomial r over F so that p = r(q 1 , q 2 , . . . , qt). 24 E.g. have*small*arithmetic circuits or*formulae*.*space*25 . ...##
###
Sentiment Analysis
[chapter]

2017
*
Encyclopedia of Machine Learning and Data Mining
*

Theor Comput Sci 261(1):119-156 Gold EM (1967) Language identification in

doi:10.1007/978-1-4899-7687-1_100512
fatcat:ce4yyqo2czftzcx2kbauglh3fu
*the**limit*. Inf Control 10(5):447-474 Haussler D (1987) Bias, version*spaces**and**Valiant's*learning framework. ... That is, S F .n/ D sup x2X n jF x j:*The*shattering coefficient*of*a hypothesis*class*H is used in generalization bounds as an*analogue*to*the*class's size in*the*finite case. ... Letting each ' i*and*j denote generic first-order*formula*, we can perform*the*"cross-sum"˚*of*case statements in*the*following manner: Likewise, we can perform « ,˝,*and*max operations by respectively subtracting ...##
###
Spike-Timing-Dependent Plasticity
[chapter]

2017
*
Encyclopedia of Machine Learning and Data Mining
*

Theor Comput Sci 261(1):119-156 Gold EM (1967) Language identification in

doi:10.1007/978-1-4899-7687-1_774
fatcat:2jprihjaxfbtpb3ttwuuz3u34y
*the**limit*. Inf Control 10(5):447-474 Haussler D (1987) Bias, version*spaces**and**Valiant's*learning framework. ... That is, S F .n/ D sup x2X n jF x j:*The*shattering coefficient*of*a hypothesis*class*H is used in generalization bounds as an*analogue*to*the*class's size in*the*finite case. ... Letting each ' i*and*j denote generic first-order*formula*, we can perform*the*"cross-sum"˚*of*case statements in*the*following manner: Likewise, we can perform « ,˝,*and*max operations by respectively subtracting ...
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