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| - | In this section, we summarize the PCA, PCA whitening and ZCA whitening algorithms,
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| - | and also describe how you can implement them using efficient linear algebra libraries.
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| - | First, we need to ensure that the data has (approximately) zero-mean. For natural images, we achieve this (approximately) by subtracting the mean value of each image patch.
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| - | We achieve this by computing the mean for each patch and subtracting it for each patch. In Matlab, we can do this by using
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| - | avg = mean(x, 1); % Compute the mean pixel intensity value separately for each patch. | + | |
| - | x = x - repmat(avg, size(x, 1), 1);
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| - | Next, we need to compute <math>\textstyle \Sigma = \frac{1}{m} \sum_{i=1}^m (x^{(i)})(x^{(i)})^T</math>. If you're implementing this in Matlab (or even if you're implementing this in C++, Java, etc., but have access to an efficient linear algebra library), doing it as an explicit sum is inefficient. Instead, we can compute this in one fell swoop as
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| - | sigma = x * x' / size(x, 2);
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| - | (Check the math yourself for correctness.)
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| - | Here, we assume that <math>x</math> is a data structure that contains one training example per column (so, <math>x</math> is a <math>\textstyle n</math>-by-<math>\textstyle m</math> matrix).
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| - | Next, PCA computes the eigenvectors of <math>\Sigma</math>. One could do this using the Matlab <tt>eig</tt> function. However, because <math>\Sigma</math> is a symmetric positive semi-definite matrix, it is more numerically reliable to do this using the <tt>svd</tt> function. Concretely, if you implement
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| - | [U,S,V] = svd(sigma);
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| - | then the matrix <math>U</math> will contain the eigenvectors of <math>Sigma</math> (one eigenvector per column, sorted in order from top to bottom eigenvector), and the diagonal entries of the matrix <math>S</math> will contain the corresponding eigenvalues (also sorted in decreasing order). The matrix <math>V</math> will be equal to transpose of <math>U</math>, and can be safely ignored.
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| - | (Note: The <tt>svd</tt> function actually computes the singular vectors and singular values of a matrix, which for the special case of a symmetric positive semi-definite matrix---which is all that we're concerned with here---is equal to its eigenvectors and eigenvalues. A full discussion of singular vectors vs. eigenvectors is beyond the scope of these notes.)
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| - | Finally, you can compute <math>\textstyle x_{\rm rot}</math> and <math>\textstyle \tilde{x}</math> as follows:
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| - | xRot = U' * x; % rotated version of the data.
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| - | xTilde = U(:,1:k)' * x; % reduced dimension representation of the data,
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| - | % where k is the number of eigenvectors to keep
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| - | This gives your PCA representation of the data in terms of <math>\textstyle \tilde{x} \in \Re^k</math>.
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| - | Incidentally, if <math>x</math> is a <math>\textstyle n</math>-by-<math>\textstyle m</math> matrix containing all your training data, this is a vectorized
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| - | implementation, and the expressions
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| - | above work too for computing <math>x_{\rm rot}</math> and <math>\tilde{x}</math> for your entire training set
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| - | all in one go. The resulting
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| - | <math>x_{\rm rot}</math> and <math>\tilde{x}</math> will have one column corresponding to each training example.
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| - | To compute the PCA whitened data <math>\textstyle x_{\rm PCAwhite}</math>, use
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| - | xPCAwhite = diag(1./sqrt(diag(S) + epsilon)) * U' * x;
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| - | Since <math>S</math>'s diagonal contains the eigenvalues <math>\textstyle \lambda_i</math>,
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| - | this turns out to be a compact way
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| - | of computing <math>\textstyle x_{{\rm PCAwhite},i} = \frac{x_{{\rm rot},i} }{\sqrt{\lambda_i}}</math>
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| - | simultaneously for all <math>\textstyle i</math>.
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| - | Finally, you can also compute the ZCA whitened data <math>\textstyle x_{\rm ZCAwhite}</math> as:
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| - | xZCAwhite = U * diag(1./sqrt(diag(S) + epsilon)) * U' * x;
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| - | {{PCA}}
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