Neural Network Vectorization

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== Sparse autoencoder ==
== Sparse autoencoder ==
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The [http://ufldl.stanford.edu/wiki/index.php/Autoencoders_and_Sparsity sparse autoencoder] neural network has an additional sparsity penalty that constrains neurons' average firing rate to be close to some target activation <math>\rho</math>. We take into the account the sparsity penalty by computing the following:
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The [[Autoencoders_and_Sparsity|sparse autoencoder]] neural network has an additional sparsity penalty that constrains neurons' average firing rate to be close to some target activation <math>\rho</math>. When performing backpropagation on a single training example, we had taken into the account the sparsity penalty by computing the following:
:<math>\begin{align}
:<math>\begin{align}
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\end{align}</math>
\end{align}</math>
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In the ''unvectorized'' case, this is computed as:
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In the ''unvectorized'' case, this was computed as:
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Recall that when we vectorizing the gradient computations, <tt>delta2</tt> is now a matrix with <math>m</math> columns corresponding to the <math>m</math> training examples.  Furthermore, notice that the <tt>sparsity_delta</tt> term is the same regardless of the example we are processing.  This suggests that vectorizing the computation above can be done by simply adding the same value to to each column when constructing the <tt>delta2</tt> matrix. Thus, to vectorize the above computations, we can simply add <tt>sparsity_delta</tt> (e.g., using <tt>repmat</tt>) to <tt>delta2</tt>.
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The code above still had a <tt>for</tt> loop over the training set, and <tt>delta2</tt> was a column vector.
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In contrast, recall that in the vectorized case, <tt>delta2</tt> is now a matrix with <math>m</math> columns corresponding to the <math>m</math> training examples.  Now, notice that the <tt>sparsity_delta</tt> term is the same regardless of what example we are processing.  This suggests that vectorizing the computation above can be done by simply adding the same value to each column when constructing the <tt>delta2</tt> matrix. Thus, to vectorize the above computation, we can simply add <tt>sparsity_delta</tt> (e.g., using <tt>repmat</tt>) to each column of <tt>delta2</tt>.

Revision as of 18:56, 29 April 2011

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