T. Poggio and F. Girosi. Regularization algorithms for learning that are equivalent to multilayer perceptrons. Science, vol. 247:978--982, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... networks are called Non Recurrent Neural Networks (NRNNs) in this work) This is a well plodded area where existing theory helps to understand the generalization properties of these architectures, and ensures the convergence of the training processes towards the desired outputs [5] 6] 7] 8] [9], t0] tt] 12] 13] 14] 15] On the other hand, the spario temporal objectives inherent to dynamical problems require neural networks with recurrent connections (referred to as Recurrent Neural Networks (RNNs) in this work) Although considerable effort has been spent in the recent past ....

....and it is therefore implementable using a NRNN. Now, given a set of data pairs (ti, y(ti) i [1, l] the output of the mapping y(t) g(o(t) gl( R R , can be made arbitrarily close to y(t) provided that g( is implemented with a NRNN with an appropriate VC dimension [5] 6] 7] 8] [9], 10] and l 00. 2 Hence, as suggested by the theorem, any bounded input bounded output trajectory generation process can always be reformulated as two cascaded processes: a first one, h( that maps time t into an intermediate trajectory o(t) and a second one, g( that maps this ....

T. Poggio and F. Girosi, "Regularization algorithms for learning that are equivalent to multilayer networks," Science, vol. 247, pp. 978-982, 1990.

....rules describing each of the constituent networks. The nature of these methods is quite di erent depending upon what type of transfer function the units in the network employ. Craven [48] discusses methods for networks that use sigmoidal transfer function and those that use local basis functions [21, 46] instead of sigmoids. Local basis functions are so named because they are designed to respond to localized patterns in their input space. There are a number of local rule extraction methods for networks that use sigmoidal transfer function for their hidden and output units. In these methods, the ....

Pggio T. and Girosi F. Regularization algorithms for learning that are equivalent to multilayer networks. Science, 247:978-982, 1990.

....Theorem was demonstrated that guarantees that an FBF network can perform approximation of continuous function at any assigned precision. As is well known, similar results on function approximation have been obtained by other feedforward connectionist systems, such as MLP s and RBF networks [4, 19]. The FBF network can be identified both by exploiting the linguistic knowledge available (structure identification problem) and by using the information contained in a data set (pa rameter estimation problem) 13] For the FBF network, the parameter estimation problem can be solved by ....

Poggio, T., and Girosi, F. "Regularization algorithms for learning that are equivalent to multilayer networks." Science, 247, 1990, pp. 978--982.

....tractability, radial basis function networks have recently also attracted consid erable attention especially in applications dealing with prediction and classification [32] 34] The importance of radial basis function networks has also greatly benefitted from the work of Poggio et.al. 35] [37], where the relationship between regularization theory and radial basis function networks is explored. The approximation capabilities of static sigmoidal type networks and of radial basis func tion networks has been studied by several research groups (see for example, 38] 40] In Section 2 we ....

T. Poggio and F. Girosi "Regularization algorithms for learning that are equivalent to multilayer networks", Sciece, vol. 247, pp. 978-982, 1990.

....x(see equation 2) Kernel bsed classifiers, hve decision function defined s: P (x) x, x) The coeciens nd he bis b re he prmeers o be djused nd he x re he rining perns. The function K is predefined kernel, for example potential function [6] or my Rdil Bsis Function (see for insurance [7]) Percepttons nd RBFs re often considered wo very disinc pproches o classification. However, for number of rining lgorihms, he resulting decision function cn be cs either in he form of equation (5) or (6) This hs been pointed ou in he literature for he Perceptton nd potential function lgorihms ....

.... lgorihms, he resulting decision function cn be cs either in he form of equation (5) or (6) This hs been pointed ou in he literature for he Perceptton nd potential function lgorihms [6] for he polynomial classifiers rined wih pseudo inverse [8] nd more recently for regulriz ion lgorihms nd RBF s [7]. In hose cses, Percepttons nd RBFs constitute dal representations of he sme decision function. The duality principle cn be understood simply in he cse of Hebb s learning rule. The weigh vector of linear Perceptton ( x) z) rined wih Hebb s rule, is simply he verge of ll rining perns x, ....

T. Poggio and F. Girosi. Regularization algorithms for learning that are equiv- alent to multilayer networks. Science, 247:978 982, February 1990.

....to biological neural networks, that do not follow this model. In particular, networks whose representations are formed around prototypes or kernels, acting as (multiple) averages of data from the task in question, are a popular approach and have been extensively studied [Kohonen, 1988a, Girosi Poggio, 1990]. Compared to the Perceptron based approach, these methods typically require more parameters (they are closely related to memory based approaches such as the classic Parzen window approach to density estimation) but have the advantage of rapid training time: it is usually easy to obtain a ....

....including the size of the available training set (typically, the smaller the training set, the greater the need for smoothing) and on the model structure. There is a large body of theoretical work concerning this issue. Smoothness, as referred to here, can be related to work on regularization [Poggio Girosi, 1990, Girosi Poggio, 1990] in which a term that acts as a penalty to an overly close fit to the training data is used during optimization. This can help the model generalize better to unseen data. One approach to choosing the degree of smoothness is cross validation [Stone, 1974] In this ....

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Poggio, T. and Girosi, F. (1990). Regularization Algorithms for Learning that are Equivalent to Multi-Layer Networks. Science, Vol. 247, pp. 978-982.

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T. Poggio and F. Girosi, Regularization algorithms for learning that are equivalent to multilayer networks, Science 247 (1990) 978--982.

....to maximizing , which is in turn equivalent to minimizing which corresponds to minimizing the RKHS norm. The regularization algorithm and learning theory The Mercer theorem was introduced in learning theory by Vapnik and RKHS by Girosi [22] and later by Vapnik [9, 50] Poggio and Girosi [41, 40, 23] had introduced Tikhonov regularization in learning theory (the reformulation of Support Vector Machines as a special case of regularization can be found in [19] Earlier, Gaussian Radial Basis Functions were proposed as an alternative to neural networks by Broomhead and Loewe. Of course, RKHS ....

T. Poggio and F. Girosi. Regularization algorithms for learning that are equivalent to multilayer networks. Science, 247:978--982, 1990.

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T. Poggio and F. Girosi. Regularization algorithms for learning that are equivalenttomultilayer networks. Science, 247:978--982, 1990.

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T. Poggio and F. Girosi. Regularization algorithms for learning that are equivalenttomultilayer networks. Science, 247:978--982, 1990b.

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T. Poggio and F. Girosi. Regularization algorithms for learning that are equivalent to multilayer perceptrons. Science, vol. 247:978--982, 1990.

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T. Poggio and F. Girosi. Regularization algorithms for learning that are equivalent to multilayer networks. Science, 247:978--982, 1990.

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Tomaso Poggio and Frederico Girosi. Regularization algorithms for learning that are equivalent to multilayer networks. Science, 247:978--982, 1990.

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Poggio, R., & Girosi, F. (1990). Regularization algorithms for learning that are equivalent to multilayer networks. Science, 247, 213--225.

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Poggio,R. & Girosi,F, "Regularization algorithms for learning that are equivalent to multilayer networks" Science Vol.247 (1990).

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T. Poggio and F. Girosi, "Regularization algorithms for learning that are equivalent to multilayer networks," Science, vol. 247, pp. 978--982, 1990.

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T. Poggio and F. Girosi, "Regularization algorithms for learning that are equivalent to multilayer network," Science, 247 (1990), pp. 978--982.

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T. Poggio and F. Girosi, "Regularization algorithms for learning that are equivalent to multilayer networks," Science, vol. 247, pp. 978--982, 1990.

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T. Poggio and F. Girosi, "Regularization algorithms for learning that are equivalent to multilayer network," Science, 247 (1990), pp. 978--982.

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Poggio, T., Girosi, F., "Regularization Algorithms for Learning That are Equivalent to Multilayer Networks," Science, vol. 247, pp. 978-982, 1990.

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R. Poggio and F. Girosi. Regularization algorithms for learning that are equivalent to multilayer networks. Science, 247:978--982, 1990.

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T. Poggio and F. Girosi. Regularization algorithms for learning that are equivalent to multilayer networks. Science, 247:978--982, 1990.

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T.Poggio and F.Girosi, , Regularization algorithms for learning that are equivalent to multilayer networks, Science, 247:pp.978-982, 1990.

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T. Poggio and F. Girosi, "Regularization algorithms for learning that are equivalent to multilayer networks," Science, vol. 247, pp. 978--982, 1990.

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Poggio T, Girosi F (1990) Regularization algorithms for learning that are equivalent to multilayer networks. Science, 247:978 982

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