mhjensen
214 lines
9.6 KiB
HTML
214 lines
9.6 KiB
HTML
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<!-- navigation toc: --> <li><a href="._NeuralNet-bs001.html#___sec0" style="font-size: 80%;"><b>Neural networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._NeuralNet-bs007.html#___sec6" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
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<!-- navigation toc: --> <li><a href="#___sec7" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
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<!-- navigation toc: --> <li><a href="._NeuralNet-bs014.html#___sec13" style="font-size: 80%;"> Matrix-vector notation</a></li>
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<!-- navigation toc: --> <li><a href="._NeuralNet-bs018.html#___sec17" style="font-size: 80%;"> Relevance</a></li>
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<!-- navigation toc: --> <li><a href="._NeuralNet-bs019.html#___sec18" style="font-size: 80%;"><b>Setting up a Multi-layer perceptron model</b></a></li>
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<h2 id="___sec7" class="anchor">Why multilayer perceptrons? </h2>
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<p>
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According to the <em>Universal approximation theorem</em>, a feed-forward neural network with just a single hidden layer containing
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a finite number of neurons can approximate a continuous multidimensional function to arbitrary accuracy,
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assuming the activation function for the hidden layer is a <b>non-constant, bounded and monotonically-increasing continuous function</b>.
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<p>
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Note that the requirements on the activation function only applies to
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the hidden layer, the output nodes are always assumed to be linear, so
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as to not restrict the range of output values.
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<p>
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We note that this theorem is only applicable to an NN with <em>one</em> hidden
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layer. Therefore, we can easily construct an NN that employs
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activation functions which do not satisfy the above requirements, as
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long as we have at least one layer with activation functions that
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<em>do</em>. Furthermore, although the universal approximation theorem lays
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the theoretical foundation for regression with neural networks, it
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does not say anything about how things work in practice: A neural
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network can still be able to approximate a given function reasonably
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well without having the flexibility to fit <em>all other</em> functions.
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