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Arxiv:1906.06684V1 [Math.NA]
Randomized Computation of Continuous Data: Is Brownian Motion Computable?∗ Willem L. Fouch´e1, Hyunwoo Lee2, Donghyun Lim2, Sewon Park2, Matthias Schr¨oder3, Martin Ziegler2 1 University of South Africa 2 KAIST 3 University of Birmingham Abstract. We consider randomized computation of continuous data in the sense of Computable Analysis. Our first contribution formally confirms that it is no loss of generality to take as sample space the Cantor space of infinite fair coin flips. This extends [Schr¨oder&Simpson’05] and [Hoyrup&Rojas’09] considering sequences of suitably and adaptively biased coins. Our second contribution is concerned with 1D Brownian Motion (aka Wiener Process), a prob- ability distribution on the space of continuous functions f : [0, 1] → R with f(0) = 0 whose computability has been conjectured [Davie&Fouch´e’2013; arXiv:1409.4667,§6]. We establish that this (higher-type) random variable is computable iff some/every computable family of moduli of continuity (as ordinary random variables) has a computable probability distribution with respect to the Wiener Measure. 1 Introduction Randomization is a powerful technique in classical (i.e. discrete) Computer Science: Many difficult problems have turned out to admit simple solutions by algorithms that ‘roll dice’ and are efficient/correct/optimal with high probability [DKM+94,BMadHS99,CS00,BV04]. Indeed, fair coin flips have been shown computationally universal [Wal77]. Over continuous data, well-known closely connected to topology [Grz57] [Wei00, 2.2+ 3], notions of proba- § § bilistic computation are more subtle [BGH15,Col15]. 1.1 Overview Section 2 resumes from [SS06] the question of how to represent Borel probability measures. -
A Relativistic Extension of Hopfield Neural Networks Via the Mechanical Analogy
A relativistic extension of Hopfield neural networks via the mechanical analogy Adriano Barra,a;b;c Matteo Beccariaa;b and Alberto Fachechia;b aDipartimento di Matematica e Fisica Ennio De Giorgi, Universit`adel Salento, Lecce, Italy bINFN, Istituto Nazionale di Fisica Nucleare, Sezione di Lecce, Italy cGNFM-INdAM, Gruppo Nazionale per la Fisica Matematica, Sezione di Lecce, Italy E-mail: [email protected], [email protected], [email protected] Abstract: We propose a modification of the cost function of the Hopfield model whose salient features shine in its Taylor expansion and result in more than pairwise interactions with alternate signs, suggesting a unified framework for handling both with deep learning and network pruning. In our analysis, we heavily rely on the Hamilton-Jacobi correspon- dence relating the statistical model with a mechanical system. In this picture, our model is nothing but the relativistic extension of the original Hopfield model (whose cost function is a quadratic form in the Mattis magnetization which mimics the non-relativistic Hamiltonian for a free particle). We focus on the low-storage regime and solve the model analytically by taking advantage of the mechanical analogy, thus obtaining a complete characterization of the free energy and the associated self-consistency equations in the thermodynamic limit. On the numerical side, we test the performances of our proposal with MC simulations, showing that the stability of spurious states (limiting the capabilities of the standard Heb- bian -
Markov Model Checking of Probabilistic Boolean Networks Representations of Genes
Markov Model Checking of Probabilistic Boolean Networks Representations of Genes Marie Lluberes1, Jaime Seguel2 and Jaime Ramírez-Vick3 1, 2 Electrical and Computer Engineering Department, University of Puerto Rico, Mayagüez, Puerto Rico 3 General Engineering Department, University of Puerto Rico, Mayagüez, Puerto Rico or, on contrary, to prevent or to stop an undesirable Abstract - Our goal is to develop an algorithm for the behavior. This “guiding” of the network dynamics is automated study of the dynamics of Probabilistic Boolean referred to as intervention. The power to intervene with the Network (PBN) representation of genes. Model checking is network dynamics has a significant impact in diagnostics an automated method for the verification of properties on and drug design. systems. Continuous Stochastic Logic (CSL), an extension of Biological phenomena manifest in the continuous-time Computation Tree Logic (CTL), is a model-checking tool domain. But, in describing such phenomena we usually that can be used to specify measures for Continuous-time employ a binary language, for instance, expressed or not Markov Chains (CTMC). Thus, as PBNs can be analyzed in expressed; on or off; up or down regulated. Studies the context of Markov theory, the use of CSL as a method for conducted restricting genes expression to only two levels (0 model checking PBNs could be a powerful tool for the or 1) suggested that information retained by these when simulation of gene network dynamics. Particularly, we are binarized is meaningful to the extent that it is remains in a interested in the subject of intervention. This refers to the continuous domain [2]. -
Online Spectral Clustering on Network Streams Yi
Online Spectral Clustering on Network Streams By Yi Jia Submitted to the graduate degree program in Electrical Engineering and Computer Science and the Graduate Faculty of the University of Kansas in partial fulfillment of the requirements for the degree of Doctor of Philosophy Jun Huan, Chairperson Swapan Chakrabarti Committee members Jerzy Grzymala-Busse Bo Luo Alfred Tat-Kei Ho Date defended: The Dissertation Committee for Yi Jia certifies that this is the approved version of the following dissertation : Online Spectral Clustering on Network Streams Jun Huan, Chairperson Date approved: ii Abstract Graph is an extremely useful representation of a wide variety of practical systems in data analysis. Recently, with the fast accumulation of stream data from various type of networks, significant research interests have arisen on spectral clustering for network streams (or evolving networks). Compared with the general spectral clustering problem, the data analysis of this new type of problems may have additional requirements, such as short processing time, scalability in distributed computing environments, and temporal variation tracking. However, to design a spectral clustering method to satisfy these requirement cer- tainly presents non-trivial efforts. There are three major challenges for the new algorithm design. The first challenge is online clustering computation. Most of the existing spectral methods on evolving networks are off-line methods, using standard eigensystem solvers such as the Lanczos method. It needs to recompute solutions from scratch at each time point. The second challenge is the paralleliza- tion of algorithms. To parallelize such algorithms is non-trivial since standard eigen solvers are iterative algorithms and the number of iterations can not be pre- determined. -
PDF with Red/Green Hyperlinks
Elements of Programming Elements of Programming Alexander Stepanov Paul McJones (ab)c = a(bc) Semigroup Press Palo Alto • Mountain View Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book, and the publisher was aware of a trademark claim, the designations have been printed with initial capital letters or in all capitals. The authors and publisher have taken care in the preparation of this book, but make no expressed or implied warranty of any kind and assume no responsibility for errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of the use of the information or programs contained herein. Copyright c 2009 Pearson Education, Inc. Portions Copyright c 2019 Alexander Stepanov and Paul McJones All rights reserved. Printed in the United States of America. This publication is protected by copyright, and permission must be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permissions, request forms and the appropriate contacts within the Pearson Education Global Rights & Permissions Department, please visit www.pearsoned.com/permissions/. ISBN-13: 978-0-578-22214-1 First printing, June 2019 Contents Preface to Authors' Edition ix Preface xi 1 Foundations 1 1.1 Categories of Ideas: Entity, Species, -
Optimization by Mean Field Annealing
91 OPTIMIZATION BY MEAN FIELD ANNEALING Griff Bilbro Reinhold Mann Thomas K. Miller ECE Dept. Eng. Physics and Math. Div. ECE Dept. NCSU Oak Ridge N atl. Lab. NCSU Raleigh, NC 27695 Oak Ridge, TN 37831 Raleigh, N C 27695 Wesley. E. Snyder David E. Van den Bout Mark White ECE Dept. ECE Dept. ECE Dept. NCSU NCSU NCSU Raleigh, NC 27695 Raleigh, NC 27695 Raleigh, NC 27695 ABSTRACT Nearly optimal solutions to many combinatorial problems can be found using stochastic simulated annealing. This paper extends the concept of simulated annealing from its original formulation as a Markov process to a new formulation based on mean field theory. Mean field annealing essentially replaces the discrete de grees of freedom in simulated annealing with their average values as computed by the mean field approximation. The net result is that equilibrium at a given temperature is achieved 1-2 orders of magnitude faster than with simulated annealing. A general frame work for the mean field annealing algorithm is derived, and its re lationship to Hopfield networks is shown. The behavior of MFA is examined both analytically and experimentally for a generic combi natorial optimization problem: graph bipartitioning. This analysis indicates the presence of critical temperatures which could be im portant in improving the performance of neural networks. STOCHASTIC VERSUS MEAN FIELD In combinatorial optimization problems, an objective function or Hamiltonian, H(s), is presented which depends on a vector of interacting 3pim, S = {81," .,8N}, in some complex nonlinear way. Stochastic simulated annealing (SSA) (S. Kirk patrick, C. Gelatt, and M. Vecchi (1983)) finds a global minimum of H by com bining gradient descent with a random process. -
7Network Models
7 Network Models 7.1 Introduction Extensive synaptic connectivity is a hallmark of neural circuitry. For ex- ample, a typical neuron in the mammalian neocortex receives thousands of synaptic inputs. Network models allow us to explore the computational potential of such connectivity, using both analysis and simulations. As illustrations, we study in this chapter how networks can perform the fol- lowing tasks: coordinate transformations needed in visually guided reach- ing, selective amplification leading to models of simple and complex cells in primary visual cortex, integration as a model of short-term memory, noise reduction, input selection, gain modulation, and associative mem- ory. Networks that undergo oscillations are also analyzed, with applica- tion to the olfactory bulb. Finally, we discuss network models based on stochastic rather than deterministic dynamics, using the Boltzmann ma- chine as an example. Neocortical circuits are a major focus of our discussion. In the neocor- tex, which forms the convoluted outer surface of the (for example) human brain, neurons lie in six vertical layers highly coupled within cylindrical columns. Such columns have been suggested as basic functional units, cortical columns and stereotypical patterns of connections both within a column and be- tween columns are repeated across cortex. There are three main classes of interconnections within cortex, and in other areas of the brain as well. Feedforward connections bring input to a given region from another re- feedforward, gion located at an earlier stage along a particular processing pathway. Re- recurrent, current synapses interconnect neurons within a particular region that are and top-down considered to be at the same stage along the processing pathway. -
Inference of a Probabilistic Boolean Network from a Single Observed Temporal Sequence
Hindawi Publishing Corporation EURASIP Journal on Bioinformatics and Systems Biology Volume 2007, Article ID 32454, 15 pages doi:10.1155/2007/32454 Research Article Inference of a Probabilistic Boolean Network from a Single Observed Temporal Sequence Stephen Marshall,1 Le Yu,1 Yufei Xiao,2 and Edward R. Dougherty2, 3, 4 1 Department of Electronic and Electrical Engineering, Faculty of Engineering, University of Strathclyde, Glasgow, G1 1XW, UK 2 Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843-3128, USA 3 Computational Biology Division, Translational Genomics Research Institute, Phoenix, AZ 85004, USA 4 Department of Pathology, University of Texas M. D. Anderson Cancer Center, Houston, TX 77030, USA Received 10 July 2006; Revised 29 January 2007; Accepted 26 February 2007 Recommended by Tatsuya Akutsu The inference of gene regulatory networks is a key issue for genomic signal processing. This paper addresses the inference of proba- bilistic Boolean networks (PBNs) from observed temporal sequences of network states. Since a PBN is composed of a finite number of Boolean networks, a basic observation is that the characteristics of a single Boolean network without perturbation may be de- termined by its pairwise transitions. Because the network function is fixed and there are no perturbations, a given state will always be followed by a unique state at the succeeding time point. Thus, a transition counting matrix compiled over a data sequence will be sparse and contain only one entry per line. If the network also has perturbations, with small perturbation probability, then the transition counting matrix would have some insignificant nonzero entries replacing some (or all) of the zeros. -
Undecidability of the Word Problem for Groups: the Point of View of Rewriting Theory
Universita` degli Studi Roma Tre Facolta` di Scienze M.F.N. Corso di Laurea in Matematica Tesi di Laurea in Matematica Undecidability of the word problem for groups: the point of view of rewriting theory Candidato Relatori Matteo Acclavio Prof. Y. Lafont ..................................... Prof. L. Tortora de falco ...................................... questa tesi ´estata redatta nell'ambito del Curriculum Binazionale di Laurea Magistrale in Logica, con il sostegno dell'Universit´aItalo-Francese (programma Vinci 2009) Anno Accademico 2011-2012 Ottobre 2012 Classificazione AMS: Parole chiave: \There once was a king, Sitting on the sofa, He said to his maid, Tell me a story, And the maid began: There once was a king, Sitting on the sofa, He said to his maid, Tell me a story, And the maid began: There once was a king, Sitting on the sofa, He said to his maid, Tell me a story, And the maid began: There once was a king, Sitting on the sofa, . " Italian nursery rhyme Even if you don't know this tale, it's easy to understand that this could continue indefinitely, but it doesn't have to. If now we want to know if the nar- ration will finish, this question is what is called an undecidable problem: we'll need to listen the tale until it will finish, but even if it will not, one can never say it won't stop since it could finish later. those things make some people loose sleep, but usually children, bored, fall asleep. More precisely a decision problem is given by a question regarding some data that admit a negative or positive answer, for example: \is the integer number n odd?" or \ does the story of the king on the sofa admit an happy ending?". -
CS411-2015F-14 Counter Machines & Recursive Functions
Automata Theory CS411-2015F-14 Counter Machines & Recursive Functions David Galles Department of Computer Science University of San Francisco 14-0: Counter Machines Give a Non-Deterministic Finite Automata a counter Increment the counter Decrement the counter Check to see if the counter is zero 14-1: Counter Machines A Counter Machine M = (K, Σ, ∆,s,F ) K is a set of states Σ is the input alphabet s ∈ K is the start state F ⊂ K are Final states ∆ ⊆ ((K × (Σ ∪ ǫ) ×{zero, ¬zero}) × (K × {−1, 0, +1})) Accept if you reach the end of the string, end in an accept state, and have an empty counter. 14-2: Counter Machines Give a Non-Deterministic Finite Automata a counter Increment the counter Decrement the counter Check to see if the counter is zero Do we have more power than a standard NFA? 14-3: Counter Machines Give a counter machine for the language anbn 14-4: Counter Machines Give a counter machine for the language anbn (a,zero,+1) (a,~zero,+1) (b,~zero,−1) (b,~zero,−1) 0 1 14-5: Counter Machines Give a 2-counter machine for the language anbncn Straightforward extension – examine (and change) two counters instead of one. 14-6: Counter Machines Give a 2-counter machine for the language anbncn (a,zero,zero,+1,0) (a,~zero,zero,+1,0) (b,~zero,~zero,-1,+1) (b,~zero,zero,−1,+1) 0 1 (c,zero,~zero,0,-1) 2 (c,zero,~zero,0,-1) 14-7: Counter Machines Our counter machines only accept if the counter is zero Does this give us any more power than a counter machine that accepts whenever the end of the string is reached in an accept state? That is, given -
Why Do Nations Obey International Law?
Review Essay Why Do Nations Obey International Law? The New Sovereignty: Compliance with InternationalRegulatory Agreements. By Abram Chayes" and Antonia Handler Chayes.*" Cambridge: Harvard University Press, 1995. Pp. xii, 404. $49.95. Fairness in International Law and Institutions. By Thomas M. Franck.- Oxford: Clarendon Press, 1995. Pp. 500. $55.00. Harold Hongju Koh Why do nations obey international law? This remains among the most perplexing questions in international relations. Nearly three decades ago, Louis Henkin asserted that "almost all nations observe almost all principles of international law and almost all of their obligations almost all of the time."' Although empirical work since then seems largely to have confirmed this hedged but optimistic description,2 scholars Felix Frankfurter Professor of Law, Emeritus, Harvard Law School ** President, Consensus Building Institute. Murray and Ida Becker Professor of Law; Director. Center for International Studtcs. New York University School of Law. t Gerard C. and Bernice Latrobe Smith Professor of International Law; Director. Orville H, Schell, Jr., Center for International Human Rights, Yale University. Thts Essay sketches arguments to be fleshed out in a forthcoming book, tentatively entitled WHY NATIONS OBEY: A THEORY OF COMPLIANCE WITH INTERNATIONAL LAW. Parts of this Review Essay derive from the 1997 \Vaynflete Lectures. Magdalen College, Oxford University, and a brief book review of the Chayeses volume in 91 Am. J. INT'L L. (forthcoming 1997). 1 am grateful to Glenn Edwards, Jessica Schafer. and Douglas Wolfe for splendid research assistance, and to Bruce Ackerman, Peter Balsam, Geoffrey Brennan. Paul David, Noah Feldman. Roger Hood, Andrew Hurrell, Mark Janis, Paul Kahn, Benedict Kingsbury, Tony Kronran. -
A Guide on Probability Distributions
powered project A guide on probability distributions R-forge distributions Core Team University Year 2008-2009 LATEXpowered Mac OS' TeXShop edited Contents Introduction 4 I Discrete distributions 6 1 Classic discrete distribution 7 2 Not so-common discrete distribution 27 II Continuous distributions 34 3 Finite support distribution 35 4 The Gaussian family 47 5 Exponential distribution and its extensions 56 6 Chi-squared's ditribution and related extensions 75 7 Student and related distributions 84 8 Pareto family 88 9 Logistic ditribution and related extensions 108 10 Extrem Value Theory distributions 111 3 4 CONTENTS III Multivariate and generalized distributions 116 11 Generalization of common distributions 117 12 Multivariate distributions 132 13 Misc 134 Conclusion 135 Bibliography 135 A Mathematical tools 138 Introduction This guide is intended to provide a quite exhaustive (at least as I can) view on probability distri- butions. It is constructed in chapters of distribution family with a section for each distribution. Each section focuses on the tryptic: definition - estimation - application. Ultimate bibles for probability distributions are Wimmer & Altmann (1999) which lists 750 univariate discrete distributions and Johnson et al. (1994) which details continuous distributions. In the appendix, we recall the basics of probability distributions as well as \common" mathe- matical functions, cf. section A.2. And for all distribution, we use the following notations • X a random variable following a given distribution, • x a realization of this random variable, • f the density function (if it exists), • F the (cumulative) distribution function, • P (X = k) the mass probability function in k, • M the moment generating function (if it exists), • G the probability generating function (if it exists), • φ the characteristic function (if it exists), Finally all graphics are done the open source statistical software R and its numerous packages available on the Comprehensive R Archive Network (CRAN∗).