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Neural Networks and Analog Computation de Hava T. Siegelmann

Neural Networks and Analog Computation: Progress in Theoretical Computer Science

Autor Hava T. Siegelmann
en Limba Engleză Hardback – dec 1998

Ne-a atras atenția modul în care Neural Networks and Analog Computation redefinește granițele dintre informatica teoretică și inteligența artificială. Abordarea propusă de Hava T. Siegelmann diferă de lucrarea Theoretical Advances in Neural Computation and Learning de Vwani Roychowdhury prin faptul că este mai puțin axată pe învățarea biologică și mult mai aplicată pe fundamentul matematic al puterii de calcul analogic. În timp ce alte titluri similare, precum Neural Networks de Raul Rojas, oferă o perspectivă generalistă, acest volum din seria Progress in Theoretical Computer Science se concentrează strict pe capabilitățile computaționale ale rețelelor neuronale sub aspectul complexității.

Notăm cu interes organizarea riguroasă a materialului. Primele capitole stabilesc cadrul teoretic prin compararea rețelelor neuronale cu modelele clasice, precum mașinile Turing probabilistice și nondeterministe. Merită menționat că progresia textului este dictată de natura „greutăților” (weights) utilizate în rețea: de la greutăți întregi la cele raționale — unde este demonstrată teorema echivalenței Turing — și culminând cu rețelele cu greutăți reale. Analiza se extinde spre concepte avansate de ierarhie computațională, explorând legătura dintre complexitatea Kolmogorov și oracolele Tally.

Structura narativă a cărții este una tehnică, unde fiecare demonstrație servește la clarificarea locului pe care rețelele neuronale îl ocupă în raport cu teza lui Church. Publicată de birkhäuser, lucrarea rămâne un punct de referință pentru cercetătorii care doresc să înțeleagă nu doar cum funcționează o rețea, ci care sunt limitele sale fundamentale de procesare în format analogic.

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Specificații

ISBN-13: 9780817639495
ISBN-10: 0817639497
Pagini: 204
Ilustrații: XIV, 181 p.
Dimensiuni: 160 x 241 x 16 mm
Greutate: 0.48 kg
Ediția:1999
Editura: birkhäuser
Colecția Progress in Theoretical Computer Science
Seria Progress in Theoretical Computer Science

Locul publicării:Boston, MA, United States

Public țintă

Research

De ce să citești această carte

Recomandăm această carte cercetătorilor și matematicienilor care doresc să aprofundeze bazele teoretice ale computației. Cititorul câștigă o înțelegere matematică riguroasă a modului în care rețelele neuronale pot depăși limitele mașinilor Turing tradiționale. Este un instrument esențial pentru a înțelege complexitatea computațională în sistemele analogice, oferind demonstrații clare pentru echivalența dintre circuitele digitale și modelele neuronale cu greutăți reale.

Descriere scurtă

Humanity's most basic intellectual quest to decipher nature and master it has led to numerous efforts to build machines that simulate the world or communi­ cate with it [Bus70, Tur36, MP43, Sha48, vN56, Sha41, Rub89, NK91, Nyc92]. The computational power and dynamic behavior of such machines is a central question for mathematicians, computer scientists, and occasionally, physicists. Our interest is in computers called artificial neural networks. In their most general framework, neural networks consist of assemblies of simple processors, or "neurons," each of which computes a scalar activation function of its input. This activation function is nonlinear, and is typically a monotonic function with bounded range, much like neural responses to input stimuli. The scalar value produced by a neuron affects other neurons, which then calculate a new scalar value of their own. This describes the dynamical behavior of parallel updates. Some of the signals originate from outside the network and act as inputs to the system, while other signals are communicated back to the environment and are thus used to encode the end result of the computation.

Cuprins

1 Computational Complexity.- 1.1 Neural Networks.- 1.2 Automata: A General Introduction.- 1.3 Finite Automata.- 1.4 The Turing Machine.- 1.5 Probabilistic Turing Machines.- 1.6 Nondeterministic Turing Machines.- 1.7 Oracle Turing Machines.- 1.8 Advice Turing Machines.- 1.9 Notes.- 2 The Model.- 2.1 Variants of the Network.- 2.2 The Network’s Computation.- 2.3 Integer Weights.- 3 Networks with Rational Weights.- 3.1 The Turing Equivalence Theorem.- 3.2 Highlights of the Proof.- 3.3 The Simulation.- 3.4 Network with Four Layers.- 3.5 Real-Time Simulation.- 3.6 Inputs and Outputs.- 3.7 Universal Network.- 3.8 Nondeterministic Computation.- 4 Networks with Real Weights.- 4.1 Simulating Circuit Families.- 4.2 Networks Simulation by Circuits.- 4.3 Networks versus Threshold Circuits.- 4.4 Corollaries.- 5 Kolmogorov Weights: Between P and P/poly.- 5.1 Kolmogorov Complexity and Reals.- 5.2 Tally Oracles and Neural Networks.- 5.3 Kolmogorov Weights and Advice Classes.- 5.4 The Hierarchy Theorem.- 6 Space and Precision.- 6.1 Equivalence of Space and Precision.- 6.2 Fixed Precision Variable Sized Nets.- 7 Universality of Sigmoidal Networks.- 7.1 Alarm Clock Machines.- 7.2 Restless Counters.- 7.3 Sigmoidal Networks are Universal.- 7.4 Conclusions.- 8 Different-limits Networks.- 8.1 At Least Finite Automata.- 8.2 Proof of the Interpolation Lemma.- 9 Stochastic Dynamics.- 9.1 Stochastic Networks.- 9.2 The Main Results.- 9.3 Integer Stochastic Networks.- 9.4 Rational Stochastic Networks.- 9.5 Real Stochastic Networks.- 9.6 Unreliable Networks.- 9.7 Nondeterministic Stochastic Networks.- 10 Generalized Processor Networks.- 10.1 Generalized Networks: Definition.- 10.2 Bounded Precision.- 10.3 Equivalence with Neural Networks.- 10.4 Robustness.- 11 Analog Computation.- 11.1 DiscreteTime Models.- 11.2 Continuous Time Models.- 11.3 Hybrid Models.- 11.4 Dissipative Models.- 12 Computation Beyond the Turing Limit.- 12.1 The Analog Shift Map.- 12.2 Analog Shift and Computation.- 12.3 Physical Relevance.- 12.4 Conclusions.

Recenzii

"All of the three primary questions are considered: What computational models can the net simulate (within polynomial bounds)? What are the computational complexity classes that are relevant to the net? How does the net (which, after all, is an analog device) relate to Church’s thesis? Moreover the power of the basic model is also analyzed when the domain of reals is replaced by the rationals and the integers."
—Mathematical Reviews
"Siegelmann's book focuses on the computational complexities of neural networks and making this research accessible...the book accomplishes the said task nicely."
---SIAM Review, Vol. 42, No 3.

Textul de pe ultima copertă

The theoretical foundations of Neural Networks and Analog Computation conceptualize neural networks as a particular type of computer consisting of multiple assemblies of basic processors interconnected in an intricate structure. Examining these networks under various resource constraints reveals a continuum of computational devices, several of which coincide with well-known classical models. What emerges is a Church-Turing-like thesis, applied to the field of analog computation, which features the neural network model in place of the digital Turing machine. This new concept can serve as a point of departure for the development of alternative, supra-Turing, computational theories. On a mathematical level, the treatment of neural computations enriches the theory of computation but also explicated the computational complexity associated with biological networks, adaptive engineering tools, and related models from the fields of control theory and nonlinear dynamics.
The topics covered in this work will appeal to a wide readership from a variety of disciplines. Special care has been taken to explain the theory clearly and concisely. The first chapter review s the fundamental terms of modern computational theory from the point of view of neural networks and serves as a reference for the remainder of the book. Each of the subsequent chapters opens with introductory material and proceeds to explain the chapter’s connection to the development of the theory. Thereafter, the concept is defined in mathematical terms.
Although the notion of a neural network essentially arises from biology, many engineering applications have been found through highly idealized and simplified models of neuron behavior. Particular areas of application have been as diverse as explosives detection in airport security, signature verification, financial and medical times series prediction, vision, speech processing, robotics, nonlinear control, and signal processing. The focus inall of these models is entirely on the behavior of networks as computer.
The material in this book will be of interest to researchers in a variety of engineering and applied sciences disciplines. In addition, the work may provide the base of a graduate-level seminar in neural networks for computer science students.