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A View of Structural Complexity Theory
[chapter]

1993
*
Current Trends in Theoretical Computer Science
*

Acknowledgments Preparation

doi:10.1142/9789812794499_0034
dblp:series/wsscs/Book093
fatcat:jcrj2u7caveejeojt6w6ouzqvq
*of*this paper was supported*in*part by*the*National Science Foundation under grant CCR86-11980 and CCR89-13584. ... We believe that this is a one*of**the*important reasons to study structural concepts*in**complexity*theory.*P*1 - 1 tt (*SPARSE*),*P*2-tt (*SPARSE*), ... . (2)*P*tt (*SPARSE*) =*P*T (*SPARSE*). ... Schöning [Sc86] proved that*the*class*P*-CLOSE is precisely*the*boolean closure*of**the*union*of**the*class*P*and*the*class*of*all*sparse**sets*, and that*P*m (*SPARSE*) is properly included*in**P*-CLOSE. ...##
###
On sparse oracles separating feasible complexity classes
[chapter]

1986
*
Lecture Notes in Computer Science
*

We fully characterize

doi:10.1007/3-540-16078-7_86
fatcat:zdq6gbv5p5aufni22657noonda
*the**sparse*oracles separating NP from*P**in*worlds where*P*=NP. We display related results about coNP, E, NE, coNE, and PSPACE. ...*In*essence, we are changing our research paradigm from*the*study*of*which problems can be relativized*in*two conflicting ways to*the*study and characterization*of**the*class*of*oracles achieving a specified ... Next we dissect*the*theorem, strengthening each direction. While doing this, we note that many*complexity*properties*of**sparse**sets*degenerate into equivalence*in*worlds where*P*= NP. ...##
###
On hard instances

2000
*
Theoretical Computer Science
*

Comparing instance

doi:10.1016/s0304-3975(98)00262-x
fatcat:mrhp3hfxnjd75ikwdd322c2uxy
*complexity*to Kolmogorov*complexity*, they introduced*the*notion*of**p*-hard instances, and conjectured that every*set*not*in**P*has*p*-hard instances. ...*In**the*unbounded version, we characterize*the*classes*of*recursive enumerable resp. recursive*sets*by hard instances.*In*bounded versions, we characterize*the*class*P*. ... Acknowledgements*The*author would like to thank for*the*comments*of*two referees. ...##
###
On Reducibility to Complex or Sparse Sets

1975
*
Journal of the ACM
*

*Sets*which are efficiently reducible (

*in*Karp's sense) to arbitrarily

*complex*

*sets*are shown to be polynomial computable. ... Analogously,

*sets*efficiently reducible to arbitrarily

*sparse*

*sets*are polynomial computable. ... Similarly,

*in*Section 5, we use

*the*lemma to show that

*the*only

*sets*which are reducible

*in*Karp's sense to

*sets*

*complex*on arbitrarily

*sparse*subsets

*of*their domains are

*the*-polynomial computable

*sets*...

##
###
L-Printable Sets

1998
*
SIAM journal on computing (Print)
*

a logspace algorithm that, on input x, outputs

doi:10.1137/s0097539796300441
fatcat:ix3b3z3dlnbhboofd3lzi3au64
*the*number*of*elements*of*A that precede x*in**the*standard lexicographic ordering*of*strings), and*the**sparse**sets**in*L. ... ., 17 (1988), pp. 1193-1202] on*P*-printable*sets*, we present some simple properties*of**the*L-printable*sets*. ...*The*authors want to thank David Mix Barrington for a counterexample to a conjecture about*sparse*regular*sets*, Alan Selman for suggesting*the*tally*set*characterization*of*L-printable*sets*and Corollary ...##
###
Page 7322 of Mathematical Reviews Vol. , Issue 90M
[page]

1990
*
Mathematical Reviews
*

*The*class

*of*

*sets*with polynomial-size circuits, often de- noted

*P*/poly, plays an important role

*in*structural

*complexity*

*the*- ory, and it is known that

*P*/poly =

*P*,,(

*SPARSE*) =

*P*;(

*SPARSE*) = Py (TALLY) = ... They also showed that

*the*classes

*P*,,(

*SPARSE*) and

*P*-CLOSE are not comparable, where

*P*-CLOSE denotes

*the*Boolean closure

*of*PU

*SPARSE*. ...

##
###
Shallow sparse autoencoders versus sparse coding algorithms for image compression

2016
*
2016 IEEE International Conference on Multimedia & Expo Workshops (ICMEW)
*

Shallow

doi:10.1109/icmew.2016.7574708
dblp:conf/icmcs/DumasRG16
fatcat:lusnshfmgvfnncrbn7ujrqmzue
*sparse*autoencoders versus*sparse*coding algorithms for image compression. ABSTRACT This paper considers*the*problem*of*image compression with shallow*sparse*autoencoders. ... A performance analysis*in*terms*of*rate-distortion tradeoff and*complexity*is conducted, comparing with LARS-Lasso, Coordinate Descent (CoD) and Orthogonal Matching Pursuit (OMP). ...*In*(3), given a*set**of*images patches M x ∈ R m×*p*and W ∈ R n×m ,*the**complexity**of**the*matrix product W M x is O(nmp).*In*(3), g α requires a sorting operation*of**complexity*O(np log (np)). ...##
###
Page 642 of Mathematical Reviews Vol. , Issue 2000a
[page]

2000
*
Mathematical Reviews
*

*In*particular, Cai and Sivakumar proved

*in*

*the*mid-1990s that

*P*has

*sparse*logspace many-one complete

*sets*if and only if L=

*P*. ...

*In*particular,

*sparse*

*sets*were studied as a way

*of*getting insight into

*the*possible iso- morphism

*of*complete

*sets*for

*complexity*classes. ...

##
###
Scalable Text Classification with Sparse Generative Modeling
[chapter]

2012
*
Lecture Notes in Computer Science
*

*In*this paper we show that

*sparse*representations can be used to reduce

*the*time

*complexity*

*of*generative classifiers to build fundamentally more scalable classifiers. ... An alternative solution is using

*sparse*computation methods to fundamentally change

*the*

*complexity*

*of*

*the*processing tasks themselves. ... Replacing

*the*full

*set*

*of*labels with

*the*evaluation list results

*in*

*sparse*O(s(w)|̺|) time

*complexity*. ...

##
###
Compressive sensing based Bayesian sparse channel estimation for OFDM communication systems: high performance and low complexity
[article]

2015
*
arXiv
*
pre-print

algorithm, which can take

arXiv:1403.0192v2
fatcat:p4umq7bdcvevjbda7auvwmwemi
*the*advantage*of**sparse*structure effectively*in**the*channel as for prior information. ...*In*other words,*the*primary objective*of*these conventional methods is to catch*the*dominant channel taps without a report*of*posterior channel uncertainty. ... Fig. 2 . 2*Sparseness**of*channel vector h depends on*the*probability*p*1 . 3 . 3*of*possible position index*sets**of*channel taps as shown*in*Fig. Eqcase*of*high-dimensional broadband channels. ...##
###
On reductions to sets that avoid EXPSPACE

1995
*
Information Processing Letters
*

A s e t B is called EXPSPACE-avoiding, i f e v ery subset

doi:10.1016/0020-0190(95)00138-3
fatcat:qc6g2rmi3jb3vo6g26o63r2t5i
*of*B*in*EXPSPACE is*sparse*.*Sparse**sets*and*sets**of*high information density (called HIGH*sets**in*5]) are shown to be EXPSPACE-avoiding. ... Investigating*the**complexity*o f*sets*A*in*EXPSPACE that honestly reduce to EXPSPACE-avoiding*sets*, we show that if*the*reducibility used has a property, c a l l e d range-constructibility, then A must ...*The*rst part is a direct consequence*of*Theorem 4.5 and*the*result that C R*p*bhd (R*p*c (*SPARSE*)) implies C =*P*for any*of**the*above*complexity*classes 2]. ...##
###
Computation times of NP sets of different densities

1984
*
Theoretical Computer Science
*

*In*this paper we study

*the*computational

*complexity*

*of*

*sets*

*of*different densities

*in*NP. ... This interpretation relate5 fundamental questIons about

*the*

*complexity*

*of*our intellectual tools to basic structural problems aboLt

*P*, NP, CoNP and PSPA~T, discussed

*in*this paper. ... Clearly, every NP-printable

*set*is

*sparse*and

*in*NPnCoNP.

*The*proofs

*of*

*the*previous results yield

*the*following. Proof. If every

*sparse*

*set*

*in*NP is

*P*-printable, then clearly every such

*set*is

*in*

*P*. ...

##
###
Reducibilities on tally and sparse sets

1991
*
RAIRO - Theoretical Informatics and Applications
*

As such,

doi:10.1051/ita/1991250302931
fatcat:4bs67jtdczhb5bvfzwkkuuv47q
*sparse**sets*and tally*sets*have played important rôles*in**the*investigation*of**the*structure*of**complexity*classes and*of*polynomial time-bounded reducibilities. ... For example,*the*class*of**sets*that have polynomial-size circuits is*the*class*of**sets*that are Turing reducible*in*polynomial time to*sparse**sets*;*the*class*of**sets*with small generalized Kolmogorov*complexity*...##
###
Local congruence of chain complexes
[article]

2020
*
arXiv
*
pre-print

*The*topology

*of*these intersections is codified within a

*set*

*of*(0-2)-dimensional chain

*complexes*. ...

*The*object

*of*this paper is to transform a

*set*

*of*local chain

*complexes*to a single global

*complex*using an equivalence relation

*of*congruence

*of*cells, solving topologically

*the*numerical inaccuracies ... A subset B

*of*open

*sets*

*in*

*the*topological space T over

*the*cell

*complex*X is a basis for

*the*topology

*of*X when all other open

*sets*can be written as union

*of*elements

*of*B. ...

##
###
Efficient Hold-Out for Subset of Regressors
[chapter]

2009
*
Lecture Notes in Computer Science
*

*The*computational

*complexity*

*of*performing

*the*hold-out is O(|H| 3 + |H| 2 n), where |H| is

*the*size

*of*

*the*hold-out

*set*and n is

*the*number

*of*basis vectors. ... Further, since

*sparse*RLS can be trained

*in*O(mn 2 ) time for several regularization parameter values

*in*parallel,

*the*fast hold-out algorithm enables efficient selection

*of*

*the*optimal parameter value ... Acknowledgments This work has been supported by Tekes,

*the*Finnish Funding Agency for Technology and Innovation (grant 40020/07) and by Academy

*of*Finland (grant 128061). ...

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