15,258 research outputs found
Pseudorandomness for Approximate Counting and Sampling
We study computational procedures that use both randomness and nondeterminism. The goal of this paper is to derandomize such procedures under the weakest possible assumptions.
Our main technical contribution allows one to āboostā a given hardness assumption: We show that if there is a problem in EXP that cannot be computed by poly-size nondeterministic circuits then there is one which cannot be computed by poly-size circuits that make non-adaptive NP oracle queries. This in particular shows that the various assumptions used over the last few years by several authors to derandomize Arthur-Merlin games (i.e., show AM = NP) are in fact all equivalent.
We also define two new primitives that we regard as the natural pseudorandom objects associated with approximate counting and sampling of NP-witnesses. We use the āboostingā theorem and hashing techniques to construct these primitives using an assumption that is no stronger than that used to derandomize AM.
We observe that Cai's proof that S_2^P ā PPā(NP) and the learning algorithm of Bshouty et al. can be seen as reductions to sampling that are not probabilistic. As a consequence they can be derandomized under an assumption which is weaker than the assumption that was previously known to suffice
On optimal language compression for sets in PSPACE/poly
We show that if DTIME[2^O(n)] is not included in DSPACE[2^o(n)], then, for
every set B in PSPACE/poly, all strings x in B of length n can be represented
by a string compressed(x) of length at most log(|B^{=n}|)+O(log n), such that a
polynomial-time algorithm, given compressed(x), can distinguish x from all the
other strings in B^{=n}. Modulo the O(log n) additive term, this achieves the
information-theoretic optimum for string compression. We also observe that
optimal compression is not possible for sets more complex than PSPACE/poly
because for any time-constructible superpolynomial function t, there is a set A
computable in space t(n) such that at least one string x of length n requires
compressed(x) to be of length 2 log(|A^=n|).Comment: submitted to Theory of Computing System
Consistency of Markov chain quasi-Monte Carlo on continuous state spaces
The random numbers driving Markov chain Monte Carlo (MCMC) simulation are
usually modeled as independent U(0,1) random variables. Tribble [Markov chain
Monte Carlo algorithms using completely uniformly distributed driving sequences
(2007) Stanford Univ.] reports substantial improvements when those random
numbers are replaced by carefully balanced inputs from completely uniformly
distributed sequences. The previous theoretical justification for using
anything other than i.i.d. U(0,1) points shows consistency for estimated means,
but only applies for discrete stationary distributions. We extend those results
to some MCMC algorithms for continuous stationary distributions. The main
motivation is the search for quasi-Monte Carlo versions of MCMC. As a side
benefit, the results also establish consistency for the usual method of using
pseudo-random numbers in place of random ones.Comment: Published in at http://dx.doi.org/10.1214/10-AOS831 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Simple extractors via constructions of cryptographic pseudo-random generators
Trevisan has shown that constructions of pseudo-random generators from hard
functions (the Nisan-Wigderson approach) also produce extractors. We show that
constructions of pseudo-random generators from one-way permutations (the
Blum-Micali-Yao approach) can be used for building extractors as well. Using
this new technique we build extractors that do not use designs and
polynomial-based error-correcting codes and that are very simple and efficient.
For example, one extractor produces each output bit separately in
time. These extractors work for weak sources with min entropy , for
arbitrary constant , have seed length , and their
output length is .Comment: 21 pages, an extended abstract will appear in Proc. ICALP 2005; small
corrections, some comments and references adde
The LĆ©vyāKhintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups
Ito's construction of Markovian solutions to stochastic equations driven by a
LĆ©vy noise is extended to nonlinear distribution dependent integrands aiming at
the effective construction of linear and nonlinear Markov semigroups and the corresponding processes with a given pseudo-differential generator. It is shown that a conditionally positive integro-differential operator (of the LĆ©vy-Khintchine type) with
variable coeffcients (diffusion, drift and LĆ©vy measure) depending Lipschitz continuously on its parameters (position and/or its distribution) generates a linear or
nonlinear Markov semigroup, where the measures are metricized by the Wasserstein-Kantorovich metrics. This is a nontrivial but natural extension to general Markov
processes of a long known fact for ordinary diffusions
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