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34
Networks of Spiking Neurons: The Third Generation of Neural Network Models
 Neural Networks
, 1997
"... The computational power of formal models for networks of spiking neurons is compared with that of other neural network models based on McCulloch Pitts neurons (i.e. threshold gates) respectively sigmoidal gates. In particular it is shown that networks of spiking neurons are computationally more powe ..."
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Cited by 140 (12 self)
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The computational power of formal models for networks of spiking neurons is compared with that of other neural network models based on McCulloch Pitts neurons (i.e. threshold gates) respectively sigmoidal gates. In particular it is shown that networks of spiking neurons are computationally more powerful than these other neural network models. A concrete biologically relevant function is exhibited which can be computed by a single spiking neuron (for biologically reasonable values of its parameters), but which requires hundreds of hidden units on a sigmoidal neural net. This article does not assume prior knowledge about spiking neurons, and it contains an extensive list of references to the currently available literature on computations in networks of spiking neurons and relevant results from neurobiology. 1 Definitions and Motivations If one classifies neural network models according to their computational units, one can distinguish three different generations. The first generation i...
Nonparametric time series prediction through adaptive model selection
 Machine Learning
, 2000
"... Abstract. We consider the problem of onestep ahead prediction for time series generated by an underlying stationary stochastic process obeying the condition of absolute regularity, describing the mixing nature of process. We make use of recent results from the theory of empirical processes, and ada ..."
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Cited by 29 (0 self)
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Abstract. We consider the problem of onestep ahead prediction for time series generated by an underlying stationary stochastic process obeying the condition of absolute regularity, describing the mixing nature of process. We make use of recent results from the theory of empirical processes, and adapt the uniform convergence framework of Vapnik and Chervonenkis to the problem of time series prediction, obtaining finite sample bounds. Furthermore, by allowing both the model complexity and memory size to be adaptively determined by the data, we derive nonparametric rates of convergence through an extension of the method of structural risk minimization suggested by Vapnik. All our results are derived for general L p error measures, and apply to both exponentially and algebraically mixing processes.
On the Complexity of Computing and Learning with Multiplicative Neural Networks
 NEURAL COMPUTATION
"... In a great variety of neuron models neural inputs are combined using the summing operation. We introduce the concept of multiplicative neural networks that contain units which multiply their inputs instead of summing them and, thus, allow inputs to interact nonlinearly. The class of multiplicative n ..."
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Cited by 24 (3 self)
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In a great variety of neuron models neural inputs are combined using the summing operation. We introduce the concept of multiplicative neural networks that contain units which multiply their inputs instead of summing them and, thus, allow inputs to interact nonlinearly. The class of multiplicative neural networks comprises such widely known and well studied network types as higherorder networks and product unit networks. We investigate the complexity of computing and learning for multiplicative neural networks. In particular, we derive upper and lower bounds on the VapnikChervonenkis (VC) dimension and the pseudo dimension for various types of networks with multiplicative units. As the most general case, we consider feedforward networks consisting of product and sigmoidal units, showing that their pseudo dimension is bounded from above by a polynomial with the same order of magnitude as the currently best known bound for purely sigmoidal networks. Moreover, we show that this bound holds even in the case when the unit type, product or sigmoidal, may be learned. Crucial for these results are calculations of solution set components bounds for new network classes. As to lower bounds we construct product unit networks of fixed depth with superlinear VC dimension. For sigmoidal networks of higher order we establish polynomial bounds that, in contrast to previous results, do not involve any restriction of the network order. We further consider various classes of higherorder units, also known as sigmapi units, that are characterized by connectivity constraints. In terms of these we derive some asymptotically tight bounds.
VapnikChervonenkis Dimension of Recurrent Neural Networks
, 1997
"... Most of the work on the VapnikChervonenkis dimension of neural networks has been focused on feedforward networks. However, recurrent networks are also widely used in learning applications, in particular when time is a relevant parameter. This paper provides lower and upper bounds for the VC dimensi ..."
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Cited by 23 (5 self)
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Most of the work on the VapnikChervonenkis dimension of neural networks has been focused on feedforward networks. However, recurrent networks are also widely used in learning applications, in particular when time is a relevant parameter. This paper provides lower and upper bounds for the VC dimension of such networks. Several types of activation functions are discussed, including threshold, polynomial, piecewisepolynomial and sigmoidal functions. The bounds depend on two independent parameters: the number w of weights in the network, and the length k of the input sequence. In contrast, for feedforward networks, VC dimension bounds can be expressed as a function of w only. An important difference between recurrent and feedforward nets is that a fixed recurrent net can receive inputs of arbitrary length. Therefore we are particularly interested in the case k AE w. Ignoring multiplicative constants, the main results say roughly the following: ffl For architectures with activation oe = a...
VC Dimension of Neural Networks
 Neural Networks and Machine Learning
, 1998
"... . This paper presents a brief introduction to VapnikChervonenkis (VC) dimension, a quantity which characterizes the difficulty of distributionindependent learning. The paper establishes various elementary results, and discusses how to estimate the VC dimension in several examples of interest in ne ..."
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Cited by 19 (3 self)
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. This paper presents a brief introduction to VapnikChervonenkis (VC) dimension, a quantity which characterizes the difficulty of distributionindependent learning. The paper establishes various elementary results, and discusses how to estimate the VC dimension in several examples of interest in neural network theory. 1 Introduction In this expository paper, we present a brief introduction to the subject of computing and estimating the VC dimension of neural network architectures. We provide precise definitions and prove several basic results, discussing also how one estimates VC dimension in several examples of interest in neural network theory. We do not address the learning and estimationtheoretic applications of VC dimension. (Roughly, the VC dimension is a number which helps to quantify the difficulty when learning from examples. The sample complexity, that is, the number of "learning instances" that one must be exposed to, in order to be reasonably certain to derive accurate p...
Almost Linear VC Dimension Bounds for Piecewise Polynomial Networks
 Neural Computation
, 1998
"... We compute upper and lower bounds on the VC dimension of feedforward networks of units with piecewise polynomial activation functions. We show that if the number of layers is fixed, then the VC dimension grows as W log W , where W is the number of parameters in the network. This result stands in opp ..."
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Cited by 12 (1 self)
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We compute upper and lower bounds on the VC dimension of feedforward networks of units with piecewise polynomial activation functions. We show that if the number of layers is fixed, then the VC dimension grows as W log W , where W is the number of parameters in the network. This result stands in opposition to the case where the number of layers is unbounded, in which case the VC dimension grows as W 2 . 1 MOTIVATION The VC dimension is an important measure of the complexity of a class of binaryvalued functions, since it characterizes the amount of data required for learning in the PAC setting (see [BEHW89, Vap82]). In this paper, we establish upper and lower bounds on the VC dimension of a specific class of multilayered feedforward neural networks. Let F be the class of binaryvalued functions computed by a feedforward neural network with W weights and k computational (noninput) units, each with a piecewise polynomial activation function. Goldberg and Jerrum [GJ95] have shown that...
Why polyhedra matter in nonlinear equation solving
, 2003
"... We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in algebraic geometry or convex geometry. Highlights include the fol ..."
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Cited by 12 (4 self)
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We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in algebraic geometry or convex geometry. Highlights include the following: (1) A completely selfcontained proof of an extension of Bernstein’s Theorem. Our extension relates volumes of polytopes with the number of connected components of the complex zero set of a polynomial system, and allows any number of polynomials and/or variables. (2) A near optimal complexity bound for computing mixed area — a quantity intimately related to counting complex roots in the plane.
Statistical Learning Control of Uncertain Systems: It is better than it seems
, 1999
"... This paper answers the last question armatively, and does so byinvoking dierentversions of ..."
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Cited by 11 (9 self)
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This paper answers the last question armatively, and does so byinvoking dierentversions of