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On the Finite-Time Complexity and Practical Computation of Approximate Stationarity Concepts of Lipschitz Functions
[article]

2022
*
arXiv
*
pre-print

For NAS, Davis Drusvyatskiy (2019) showed that ρ-weakly convex

arXiv:2112.09002v2
fatcat:7dlalqfemfhznpisxwcnyqkd74
*functions*admit*finite*-time computation, while Tian So (2021) provided*the*matching impossibility results*of*dimension-free*finite*-time*complexity*... For GAS, our scheme removes*the*unrealistic subgradient selection oracle assumption in (Zhang et al., 2020, Assumption 1) and computes GAS with*the*same*finite*-time*complexity*. ... In contrast to*the*asymptotic regime,*the**finite*-time*complexity*in*the*general "non"setting is still developing. ...##
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On the algorithmic complexity of static structures

2010
*
Journal of Systems Science and Complexity
*

From

doi:10.1007/s11424-010-8465-2
fatcat:3qhipfjdyvcc7fb5tgjx474ima
*the*theory*of*algorithmic information introduced by Chaitin [4] it is known that*the*more*complex*a system that is acting*on*a*random*binary input sequence*the*more it can deform*the*stochastic properties ...*The*level*of**complexity**of**the*system is controlled via external parameters.*The*output response is*the*field*of*displacements observed at several positions*on**the*body. ... He modeled a solid as a selection rule*of*a*finite*algorithmic*complexity*which acts*on*an incoming*random*sequence*of*particles in*the*surroundings. ...##
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Page 590 of American Society of Civil Engineers. Collected Journals Vol. 110, Issue 4
[page]

1984
*
American Society of Civil Engineers. Collected Journals
*

For more

*complex*shell geometry and more*complex*correlation*function**of**the*loads which are*random*both in time and space, numerical methods may be desirable. ... Olson and Lindberg (26) and Olson (25), developed a consistent*finite*element method for analyzing*the**random*response*of**complex*struc- tures. ...##
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On the non-randomness of maximum Lempel Ziv complexity sequences of finite size

2013
*
Chaos
*

We prove that typical

doi:10.1063/1.4808251
pmid:23822483
fatcat:v6n6k54rqrdnzarsvz6arkvau4
*random*sequences*of**finite*length fall short*of**the*maximum Lempel-Ziv*complexity*, contrary to common belief. ... We discuss that, for a*finite*length, maximum Lempel-Ziv sequences can be built from a well defined generating algorithm, which makes them*of*low Kolmogorov-Chaitin*complexity*, quite*the*opposite to*randomness*... It has been proved 14 that a*finite**random*sequence with maximum KC*complexity*does not have*the*same number*of*zeros and*ones*. ...##
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Convergence of the Complex Envelope of Bandlimited OFDM Signals

2010
*
IEEE Transactions on Information Theory
*

In particular, a large amount

doi:10.1109/tit.2010.2059550
fatcat:tlkqbsbpdnfp7cxutvzttj3gye
*of*work has focused*on*analyzing*the*variation*of**the**complex*envelope*of**the*transmitted signal and*on*designing methods to minimize this variation. ... This shows that*the*properties*of**the*OFDM signal will asymptotically approach those*of*a Gaussian*random*process over any*finite*time interval. ... ACKNOWLEDGMENT*The*authors are indebted to*the*editor and reviewers for feedback that both greatly streamlined*the*proofs*of**the*original manuscript and improved*the*framing*of**the*results*of**the*paper ...##
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Computability, Complexity and Randomness (Dagstuhl Seminar 12021)

2012
*
Dagstuhl Reports
*

Research

doi:10.4230/dagrep.2.1.19
dblp:journals/dagstuhl-reports/BecherBDM12
fatcat:bkapirz4vfgkzcexiutm33k74q
*on**the*notions*of*information and*randomness*has drawn*on*methods and ideas from computability theory and cumputational*complexity*, as well as core mathematical subjects like measure theory and ... This report collects*the*material presented during*the*course*of**the*seminar. ...*The*problem posed by Kolmogorov*on*a notion*of**randomness**of**finite*objects remains unsolved. This is also*the*case for arbitrary countable objects. ...##
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Algorithmic statistics

2001
*
IEEE Transactions on Information Theory
*

While Kolmogorov

doi:10.1109/18.945257
fatcat:qe7s4mib4vff5eawevwxdrgwyu
*complexity*is*the*accepted absolute measure*of*information content*of*an individual*finite*object, a similarly absolute notion is needed for*the*relation between an individual data sample ... We also strengthen and elaborate*on*earlier results for*the*"Kolmogorov structure*function*" and "absolutely nonstochastic objects"-those objects for which*the*simplest models that summarize their relevant ... Levin for some enlightening discussions*on*Kolmogorov's "structure*function*." ...##
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Page 494 of American Mathematical Society. Proceedings of the American Mathematical Society Vol. 6, Issue 3
[page]

1955
*
American Mathematical Society. Proceedings of the American Mathematical Society
*

We can define a particular

*complex*valued*random*variable Z in*the*following way: | Z| =r with probability*one*, and arg Z is uniformly distributed*on**the*interval |[—7, 7]. ...*ON*A CLASS*OF**RANDOM*VARIABLES FRANK SPITZER 1. Introduction. ...##
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On Empirical Entropy
[article]

2011
*
arXiv
*
pre-print

Whereas previously

arXiv:1103.5985v1
fatcat:ptr7avn7bvdhbduxy3w7c3xcm4
*one*considers*the*naked entropy*of*(possibly higher order) Markov processes, we consider*the*sum*of**the*description*of**the**random*variable involved plus*the*entropy it induces. ... We propose a compression-based version*of**the*empirical entropy*of*a*finite*string over a*finite*alphabet. ... This difference parallels that between*the*Kolmogorov*complexity**of*a single*finite*object and*the*entropy*of*a*random*variable. ...##
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Prefix and plain Kolmogorov complexity characterizations of 2-randomness: simple proofs
[article]

2013
*
arXiv
*
pre-print

In [16] Miller also gave a quantitative version

arXiv:1310.5230v1
fatcat:2w26f2kpjjdr7hv3byvk3sqmh4
*of**the*first result:*the*0'-*randomness*deficiency*of*a sequence ω equals lim inf [n - C (ω1 . . . ωn)] + O(1). ... that have infinitely many initial segments with O(1)-maximal plain*complexity*(among*the*strings*of**the*same length). ... Introduction*The*connection between*complexity*and*randomness*is*one**of**the*basic ideas that motivated*the*development*of*algorithmic information theory and algorithmic*randomness*theory. ...##
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Prefix and plain Kolmogorov complexity characterizations of 2-randomness: simple proofs

2015
*
Archive for Mathematical Logic
*

In [16] Miller also gave a quantitative version

doi:10.1007/s00153-015-0430-2
fatcat:uzhs5w2u7zegpgy254t3hc3cza
*of**the*first result:*the*0 ′*randomness*deficiency*of*a sequence ω equals lim infn[n − C (ω1 . . . ωn)] + O(1). ... sequences that have infinitely many initial segments with O(1)-maximal plain*complexity*(among*the*strings*of**the*same length). ... Introduction*The*connection between*complexity*and*randomness*is*one**of**the*basic ideas that motivated*the*development*of*algorithmic information theory and algorithmic*randomness*theory. ...##
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A generalization of Levin-Schnorr's theorem
[article]

2017
*
arXiv
*
pre-print

In this paper, we will generalize

arXiv:1310.3091v2
fatcat:g4af4x5edrhdjd4ctczezwprru
*the*definition*of*partially*random*or*complex*reals, and then show*the*duality*of**random*and*complex*, i.e., a generalized version*of*Levin-Schnorr's theorem. ... We also study*randomness*from*the*view point*of*arithmetic using*the*relativization to a complete Π^0_1-class. ... It will show that these two definitions*of**randomness*have a duality, in other words, given a new notion*of**randomness*in*one**of**the*above, then,*one*can automatically get*the*definition*of**the*same notion ...##
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Finite state incompressible infinite sequences

2016
*
Information and Computation
*

*Finite*State Incompressible Infinite Sequences 2 / 21 Motivation

*The*incomputability

*of*all descriptional

*complexities*is an obstacle towards more "down-to-earth" applications

*of*AIT (e.g. for practical ...

*Finite*State Incompressible Infinite Sequences 2 / 21 Motivation

*The*incomputability

*of*all descriptional

*complexities*is an obstacle towards more "down-to-earth" applications

*of*AIT (e.g. for practical ... In particular,

*finite*state

*complexities*based

*on*some exotic enumerations behave like

*the*plain (Kolmogorov)

*complexity*. ...

##
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Symbolic complexity for nucleotide sequences: a sign of the genome structure

2016
*
Journal of Physics A: Mathematical and Theoretical
*

We introduce a method to estimate

doi:10.1088/1751-8113/49/44/445601
fatcat:ihnsaxi5mnajfjj6jsvjsuhjdq
*the**complexity**function**of*symbolic dynamical systems from a*finite*sequence*of*symbols. ... We test such*complexity*estimator*on*several symbolic dynamical systems whose*complexity**functions*are known exactly. ... It was found that*the**complexity**function*for a*finite*string has a profile which is independent*on*how*the*string was produced [4, 5] . ...##
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Beyond Physics? On the Prospects of Finding a Meaningful Oracle

2014
*
Foundations of Science
*

Physicalism would then be violated by

doi:10.1007/s10699-014-9349-z
fatcat:juqvbuwpdnb6hh6scrri67azui
*the*existence*of*oracles that produce certain kinds*of*noncomputable*functions*. ... necessity"-algorithmic rules and*randomness*. ... an earlier version*of**the*manuscript. ...
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