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

Lai Tian, Kaiwen Zhou, Anthony Man-Cho So
2022 arXiv   pre-print
For NAS, Davis Drusvyatskiy (2019) showed that ρ-weakly convex 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.  ... 
arXiv:2112.09002v2 fatcat:7dlalqfemfhznpisxwcnyqkd74

On the algorithmic complexity of static structures

Joel Ratsaby, J. Chaskalovic
2010 Journal of Systems Science and Complexity  
From 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.  ... 
doi:10.1007/s11424-010-8465-2 fatcat:3qhipfjdyvcc7fb5tgjx474ima

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.  ... 

On the non-randomness of maximum Lempel Ziv complexity sequences of finite size

E. Estevez-Rams, R. Lora Serrano, B. Aragón Fernández, I. Brito Reyes
2013 Chaos  
We prove that typical 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.  ... 
doi:10.1063/1.4808251 pmid:23822483 fatcat:v6n6k54rqrdnzarsvz6arkvau4

Convergence of the Complex Envelope of Bandlimited OFDM Signals

Shuangqing Wei, Dennis L. Goeckel, Patrick A. Kelly
2010 IEEE Transactions on Information Theory  
In particular, a large amount 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  ... 
doi:10.1109/tit.2010.2059550 fatcat:tlkqbsbpdnfp7cxutvzttj3gye

Computability, Complexity and Randomness (Dagstuhl Seminar 12021)

Veronica Becher, Laurent Bienvenu, Rodney Downey, Elvira Mayordomo, Marc Herbstritt
2012 Dagstuhl Reports  
Research 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.  ... 
doi:10.4230/dagrep.2.1.19 dblp:journals/dagstuhl-reports/BecherBDM12 fatcat:bkapirz4vfgkzcexiutm33k74q

Algorithmic statistics

P. Gacs, J.T. Tromp, P.M.B. Vitanyi
2001 IEEE Transactions on Information Theory  
While Kolmogorov 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."  ... 
doi:10.1109/18.945257 fatcat:qe7s4mib4vff5eawevwxdrgwyu

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.  ... 

On Empirical Entropy [article]

Paul M.B. Vitányi
2011 arXiv   pre-print
Whereas previously 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.  ... 
arXiv:1103.5985v1 fatcat:ptr7avn7bvdhbduxy3w7c3xcm4

Prefix and plain Kolmogorov complexity characterizations of 2-randomness: simple proofs [article]

Bruno Bauwens
2013 arXiv   pre-print
In [16] Miller also gave a quantitative version 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.  ... 
arXiv:1310.5230v1 fatcat:2w26f2kpjjdr7hv3byvk3sqmh4

Prefix and plain Kolmogorov complexity characterizations of 2-randomness: simple proofs

Bruno Bauwens
2015 Archive for Mathematical Logic  
In [16] Miller also gave a quantitative version 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.  ... 
doi:10.1007/s00153-015-0430-2 fatcat:uzhs5w2u7zegpgy254t3hc3cza

A generalization of Levin-Schnorr's theorem [article]

Keita Yokoyama
2017 arXiv   pre-print
In this paper, we will generalize 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  ... 
arXiv:1310.3091v2 fatcat:g4af4x5edrhdjd4ctczezwprru

Finite state incompressible infinite sequences

Cristian S. Calude, Ludwig Staiger, Frank Stephan
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.  ... 
doi:10.1016/j.ic.2015.11.003 fatcat:g3p6qls62nc6hl2mp4ahsntsuy

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

R Salgado-García, E Ugalde
2016 Journal of Physics A: Mathematical and Theoretical  
We introduce a method to estimate 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] .  ... 
doi:10.1088/1751-8113/49/44/445601 fatcat:ihnsaxi5mnajfjj6jsvjsuhjdq

Beyond Physics? On the Prospects of Finding a Meaningful Oracle

Taner Edis, Maarten Boudry
2014 Foundations of Science  
Physicalism would then be violated by the existence of oracles that produce certain kinds of noncomputable functions.  ...  necessity"-algorithmic rules and randomness.  ...  an earlier version of the manuscript.  ... 
doi:10.1007/s10699-014-9349-z fatcat:juqvbuwpdnb6hh6scrri67azui
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