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            <title xml:lang="en">Automatic construction of multilayer networks for non linear regression</title>
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                <forename type="first">Thierry</forename>
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                <forename type="first">Régis</forename>
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                <forename>Jean-Baptiste</forename>
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            <funder>This work has been supported by EEC funded Esprit II project nr. 5433 (NEUFODI); partners: BIKIT, ARIAI, Elorduy Sancho y Cia, LABEIN, Lyonnaise des Eaux-Dumez; Associated partner: RHEA S.A.</funder>
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            <idno type="halRefHtml">&lt;i&gt;ICANN'93&lt;/i&gt;, Sep 1993, Amsterdam, Netherlands. &lt;a target="_blank" href="https://dx.doi.org/10.1007/978-1-4471-2063-6_130"&gt;&amp;#x27E8;10.1007/978-1-4471-2063-6_130&amp;#x27E9;&lt;/a&gt;</idno>
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            <idno type="stamp" n="CNRS">CNRS - Centre national de la recherche scientifique</idno>
            <idno type="stamp" n="UNIV-COMPIEGNE">Université de Technologie de Compiègne</idno>
            <idno type="stamp" n="UNIV-TROYES">Université de Technologie de Troyes</idno>
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                <title xml:lang="en">Automatic construction of multilayer networks for non linear regression</title>
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                    <forename type="first">Thierry</forename>
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                    <forename type="first">Régis</forename>
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              <p>Although the back-propagation algorithm has proved very efficient in learning powerful internal representations, it leaves open the question of the optimal dimensionality of the space spanned by the hidden units. In [1], a novel approach to this problem has been proposed. This approach, applicable to discrimination problems, is based on direct optimization of an objective function for internal representations, which can be computed given a set of examples without specifying the network’s outputs. Coupled with a strategy for recruiting units during the learning process, this concept provides a scheme for training a multilayer network layer by layer, until a simple correspondence can be found between the final, highest-level representations and a simple target coding scheme. The objective function introduced in [1] was a measure of class separability that resulted from the analysis of the relationships between discriminant analysis and multilayer neural networks. In this paper, this approach is transposed to the approximation of continuous functions. In that case, a useful strategy consists in increasing the linear dependency between the activations in the hidden layer and the target outputs. As a common measure of linearity, the sample coefficient of multiple determination p2 is therefore a good candidate for an objective function.The constructive algorithm is basically the same as described in [1]. The initial architecture consists of N0 units in a single layer. A new randomly initialized unit is then added to the layer, and all the weights in that layer are iteratively updated so as to increase the value of the ob jective function, starting from the previous configuration. The layer is expanded until the addition of a new unit fails to increase the value of the objective function by a significant amount. The process can then be repeated with a new layer, until again no further improvement can be gained. The hidden-to-output weights can be computed in one step using a pseudo-inverse approach.Numerical experiments show that this procedure is quite robust, with very small variability from one learning curve to the next, starting from different initial conditions. This robustness, which avoids doing many trials to reach a good solution, somehow compensates for the computational burden of inverting a matrix of size equal to the number of hidden units at each iteration. Future research effort will aim at extending the search space of the algorithm to different architectures and different types of hidden nodes.</p>
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