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Current Trends in Theoretical Computer Science
Acknowledgments Preparation 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. ...doi:10.1142/9789812794499_0034 dblp:series/wsscs/Book093 fatcat:jcrj2u7caveejeojt6w6ouzqvq
Lecture Notes in Computer Science
We fully characterize 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. ...doi:10.1007/3-540-16078-7_86 fatcat:zdq6gbv5p5aufni22657noonda
Comparing instance 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. ...doi:10.1016/s0304-3975(98)00262-x fatcat:mrhp3hfxnjd75ikwdd322c2uxy
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 ...doi:10.1145/321892.321895 fatcat:n47oyw7ebjebniiyelgtqj3tea
a logspace algorithm that, on input x, outputs 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 ...doi:10.1137/s0097539796300441 fatcat:ix3b3z3dlnbhboofd3lzi3au64
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. 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)). ...doi:10.1109/icmew.2016.7574708 dblp:conf/icmcs/DumasRG16 fatcat:lusnshfmgvfnncrbn7ujrqmzue
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. ...
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. ...doi:10.1007/978-3-642-32695-0_41 fatcat:zi6k53dt4fbv3g4xpzv7djx3dq
algorithm, which can take 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. ...arXiv:1403.0192v2 fatcat:p4umq7bdcvevjbda7auvwmwemi
A s e t B is called EXPSPACE-avoiding, i f e v ery subset 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]. ...doi:10.1016/0020-0190(95)00138-3 fatcat:qc6g2rmi3jb3vo6g26o63r2t5i
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. ...doi:10.1016/0304-3975(84)90111-7 fatcat:t7z5662iyvchjbubcrgbgardva
As such, 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 ...doi:10.1051/ita/1991250302931 fatcat:4bs67jtdczhb5bvfzwkkuuv47q
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. ...arXiv:2004.00046v1 fatcat:ie57rku3jrfpnacjkkfistfdxa
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). ...doi:10.1007/978-3-642-04921-7_36 fatcat:2fhiga6pybbovpzfilhmkczvsa
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