UFLDL Tutorial_Preprocessing: PCA and Whitening

PCA

Introduction

Principal Components Analysis (PCA) is a dimensionality reduction algorithm that can be used to significantly speed up your unsupervised feature learning algorithm. More importantly, understanding PCA will enable us to later implement whitening, which is an important pre-processing step for many algorithms.

Suppose you are training your algorithm on images. Then the input will be somewhat redundant, because the values of adjacent pixels in an image are highly correlated. Concretely, suppose we are training on 16x16 grayscale image patches. Then  are 256 dimensional vectors, with one feature  corresponding to the intensity of each pixel. Because of the correlation between adjacent pixels, PCA will allow us to approximate the input with a much lower dimensional one, while incurring very little error.

Example and Mathematical Background

For our running example, we will use a dataset  with  dimensional inputs, so that . Suppose we want to reduce the data from 2 dimensions to 1. (In practice, we might want to reduce data from 256 to 50 dimensions, say; but using lower dimensional data in our example allows us to visualize the algorithms better.) Here is our dataset:

This data has already been pre-processed so that each of the features  and  have about the same mean (zero) and variance.

For the purpose of illustration, we have also colored each of the points one of three colors, depending on their  value; these colors are not used by the algorithm, and are for illustration only.

PCA will find a lower-dimensional subspace onto which to project our data. From visually examining the data, it appears that  is the principal direction of variation of the data, and  the secondary direction of variation:

I.e., the data varies much more in the direction  than . To more formally find the directions  and , we first compute the matrix  as follows:

If  has zero mean, then  is exactly the covariance matrix of . (The symbol "", pronounced "Sigma", is the standard notation for denoting the covariance matrix. Unfortunately it looks just like the summation symbol, as in ; but these are two different things.)

It can then be shown that ---the principal direction of variation of the data---is the top (principal) eigenvector of , and is the second eigenvector.

Note: If you are interested in seeing a more formal mathematical derivation/justification of this result, see the CS229 (Machine Learning) lecture notes on PCA (link at bottom of this page). You won't need to do so to follow along this course, however.

You can use standard numerical linear algebra software to find these eigenvectors (see Implementation Notes). Concretely, let us compute the eigenvectors of , and stack the eigenvectors in columns to form the matrix :

Here,  is the principal eigenvector (corresponding to the largest eigenvalue),  is the second eigenvector, and so on. Also, let be the corresponding eigenvalues.

The vectors  and  in our example form a new basis in which we can represent the data. Concretely, let  be some training example. Then  is the length (magnitude) of the projection of  onto the vector .

Similarly,  is the magnitude of  projected onto the vector .

Rotating the Data

Thus, we can represent  in the -basis by computing

(The subscript "rot" comes from the observation that this corresponds to a rotation (and possibly reflection) of the original data.) Lets take the entire training set, and compute  for every . Plotting this transformed data , we get:

This is the training set rotated into the , basis. In the general case,  will be the training set rotated into the basis ,, ...,.

One of the properties of  is that it is an "orthogonal" matrix, which means that it satisfies . So if you ever need to go from the rotated vectors  back to the original data , you can compute

because .

Reducing the Data Dimension

We see that the principal direction of variation of the data is the first dimension  of this rotated data. Thus, if we want to reduce this data to one dimension, we can set

More generally, if  and we want to reduce it to a  dimensional representation  (where ), we would take the first components of , which correspond to the top  directions of variation.

Another way of explaining PCA is that  is an  dimensional vector, where the first few components are likely to be large (e.g., in our example, we saw that  takes reasonably large values for most examples ), and the later components are likely to be small (e.g., in our example,  was more likely to be small). What PCA does it it drops the the later (smaller) components of , and just approximates them with 0's. Concretely, our definition of  can also be arrived at by using an approximation to  where all but the first  components are zeros. In other words, we have:

In our example, this gives us the following plot of  (using ):

However, since the final  components of  as defined above would always be zero, there is no need to keep these zeros around, and so we define  as a -dimensional vector with just the first  (non-zero) components.

This also explains why we wanted to express our data in the  basis: Deciding which components to keep becomes just keeping the top  components. When we do this, we also say that we are "retaining the top  PCA (or principal) components."

Recovering an Approximation of the Data

Now,  is a lower-dimensional, "compressed" representation of the original . Given , how can we recover an approximation  to the original value of ? From an earlier section, we know that . Further, we can think of  as an approximation to , where we have set the last  components to zeros. Thus, given , we can pad it out with  zeros to get our approximation to . Finally, we pre-multiply by  to get our approximation to . Concretely, we get

The final equality above comes from the definition of  given earlier. (In a practical implementation, we wouldn't actually zero pad  and then multiply by , since that would mean multiplying a lot of things by zeros; instead, we'd just multiply  with the first  columns of  as in the final expression above.) Applying this to our dataset, we get the following plot for :

We are thus using a 1 dimensional approximation to the original dataset.

If you are training an autoencoder or other unsupervised feature learning algorithm, the running time of your algorithm will depend on the dimension of the input. If you feed  into your learning algorithm instead of , then you'll be training on a lower-dimensional input, and thus your algorithm might run significantly faster. For many datasets, the lower dimensional representation can be an extremely good approximation to the original, and using PCA this way can significantly speed up your algorithm while introducing very little approximation error.

Number of components to retain

How do we set ; i.e., how many PCA components should we retain? In our simple 2 dimensional example, it seemed natural to retain 1 out of the 2 components, but for higher dimensional data, this decision is less trivial. If  is too large, then we won't be compressing the data much; in the limit of , then we're just using the original data (but rotated into a different basis). Conversely, if  is too small, then we might be using a very bad approximation to the data.

To decide how to set , we will usually look at the percentage of variance retained for different values of . Concretely, if , then we have an exact approximation to the data, and we say that 100% of the variance is retained. I.e., all of the variation of the original data is retained. Conversely, if , then we are approximating all the data with the zero vector, and thus 0% of the variance is retained.

More generally, let  be the eigenvalues of  (sorted in decreasing order), so that  is the eigenvalue corresponding to the eigenvector . Then if we retain  principal components, the percentage of variance retained is given by:

In our simple 2D example above, , and . Thus, by keeping only  principal components, we retained , or 91.3% of the variance.

A more formal definition of percentage of variance retained is beyond the scope of these notes. However, it is possible to show that . Thus, if , that shows that  is usually near 0 anyway, and we lose relatively little by approximating it with a constant 0. This also explains why we retain the top principal components (corresponding to the larger values of ) instead of the bottom ones. The top principal components  are the ones that're more variable and that take on larger values, and for which we would incur a greater approximation error if we were to set them to zero.

In the case of images, one common heuristic is to choose  so as to retain 99% of the variance. In other words, we pick the smallest value of  that satisfies

Depending on the application, if you are willing to incur some additional error, values in the 90-98% range are also sometimes used. When you describe to others how you applied PCA, saying that you chose  to retain 95% of the variance will also be a much more easily interpretable description than saying that you retained 120 (or whatever other number of) components.

PCA on Images

For PCA to work, usually we want each of the features  to have a similar range of values to the others (and to have a mean close to zero). If you've used PCA on other applications before, you may therefore have separately pre-processed each feature to have zero mean and unit variance, by separately estimating the mean and variance of each feature . However, this isn't the pre-processing that we will apply to most types of images. Specifically, suppose we are training our algorithm on natural images, so that  is the value of pixel . By "natural images," we informally mean the type of image that a typical animal or person might see over their lifetime.

Note: Usually we use images of outdoor scenes with grass, trees, etc., and cut out small (say 16x16) image patches randomly from these to train the algorithm. But in practice most feature learning algorithms are extremely robust to the exact type of image it is trained on, so most images taken with a normal camera, so long as they aren't excessively blurry or have strange artifacts, should work.

When training on natural images, it makes little sense to estimate a separate mean and variance for each pixel, because the statistics in one part of the image should (theoretically) be the same as any other. This property of images is calledstationarity.

In detail, in order for PCA to work well, informally we require that (i) The features have approximately zero mean, and (ii) The different features have similar variances to each other. With natural images, (ii) is already satisfied even without variance normalization, and so we won't perform any variance normalization. (If you are training on audio data---say, on spectrograms---or on text data---say, bag-of-word vectors---we will usually not perform variance normalization either.) In fact, PCA is invariant to the scaling of the data, and will return the same eigenvectors regardless of the scaling of the input. More formally, if you multiply each feature vector  by some positive number (thus scaling every feature in every training example by the same number), PCA's output eigenvectors will not change.

So, we won't use variance normalization. The only normalization we need to perform then is mean normalization, to ensure that the features have a mean around zero. Depending on the application, very often we are not interested in how bright the overall input image is. For example, in object recognition tasks, the overall brightness of the image doesn't affect what objects there are in the image. More formally, we are not interested in the mean intensity value of an image patch; thus, we can subtract out this value, as a form of mean normalization.

Concretely, if  are the (grayscale) intensity values of a 16x16 image patch (), we might normalize the intensity of each image  as follows:

, for all 

Note that the two steps above are done separately for each image , and that  here is the mean intensity of the image . In particular, this is not the same thing as estimating a mean value separately for each pixel .

If you are training your algorithm on images other than natural images (for example, images of handwritten characters, or images of single isolated objects centered against a white background), other types of normalization might be worth considering, and the best choice may be application dependent. But when training on natural images, using the per-image mean normalization method as given in the equations above would be a reasonable default.

References

http://cs229.stanford.edu

Whitening

Introduction

We have used PCA to reduce the dimension of the data. There is a closely related preprocessing step called whitening (or, in some other literatures, sphering) which is needed for some algorithms. If we are training on images, the raw input is redundant, since adjacent pixel values are highly correlated. The goal of whitening is to make the input less redundant; more formally, our desiderata are that our learning algorithms sees a training input where (i) the features are less correlated with each other, and (ii) the features all have the same variance.

2D example

We will first describe whitening using our previous 2D example. We will then describe how this can be combined with smoothing, and finally how to combine this with PCA.

How can we make our input features uncorrelated with each other? We had already done this when computing . Repeating our previous figure, our plot for  was:

The covariance matrix of this data is given by:

(Note: Technically, many of the statements in this section about the "covariance" will be true only if the data has zero mean. In the rest of this section, we will take this assumption as implicit in our statements. However, even if the data's mean isn't exactly zero, the intuitions we're presenting here still hold true, and so this isn't something that you should worry about.)

It is no accident that the diagonal values are  and . Further, the off-diagonal entries are zero; thus,  and  are uncorrelated, satisfying one of our desiderata for whitened data (that the features be less correlated).

To make each of our input features have unit variance, we can simply rescale each feature  by . Concretely, we define our whitened data  as follows:

Plotting , we get:

This data now has covariance equal to the identity matrix . We say that  is our PCA whitened version of the data: The different components of  are uncorrelated and have unit variance.

Whitening combined with dimensionality reduction. If you want to have data that is whitened and which is lower dimensional than the original input, you can also optionally keep only the top  components of . When we combine PCA whitening with regularization (described later), the last few components of  will be nearly zero anyway, and thus can safely be dropped.

ZCA Whitening

Finally, it turns out that this way of getting the data to have covariance identity  isn't unique. Concretely, if  is any orthogonal matrix, so that it satisfies  (less formally, if  is a rotation/reflection matrix), then will also have identity covariance. In ZCA whitening, we choose . We define

Plotting , we get:

It can be shown that out of all possible choices for , this choice of rotation causes  to be as close as possible to the original input data .

When using ZCA whitening (unlike PCA whitening), we usually keep all  dimensions of the data, and do not try to reduce its dimension.

Regularizaton

When implementing PCA whitening or ZCA whitening in practice, sometimes some of the eigenvalues  will be numerically close to 0, and thus the scaling step where we divide by  would involve dividing by a value close to zero; this may cause the data to blow up (take on large values) or otherwise be numerically unstable. In practice, we therefore implement this scaling step using a small amount of regularization, and add a small constant  to the eigenvalues before taking their square root and inverse:

When  takes values around , a value of  might be typical.

For the case of images, adding  here also has the effect of slightly smoothing (or low-pass filtering) the input image. This also has a desirable effect of removing aliasing artifacts caused by the way pixels are laid out in an image, and can improve the features learned (details are beyond the scope of these notes).

ZCA whitening is a form of pre-processing of the data that maps it from  to . It turns out that this is also a rough model of how the biological eye (the retina) processes images. Specifically, as your eye perceives images, most adjacent "pixels" in your eye will perceive very similar values, since adjacent parts of an image tend to be highly correlated in intensity. It is thus wasteful for your eye to have to transmit every pixel separately (via your optic nerve) to your brain. Instead, your retina performs a decorrelation operation (this is done via retinal neurons that compute a function called "on center, off surround/off center, on surround") which is similar to that performed by ZCA. This results in a less redundant representation of the input image, which is then transmitted to your brain.

Implementing PCA/Whitening

In this section, we summarize the PCA, PCA whitening and ZCA whitening algorithms, and also describe how you can implement them using efficient linear algebra libraries.

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.

We achieve this by computing the mean for each patch and subtracting it for each patch. In Matlab, we can do this by using

avg = mean(x, 1);     % Compute the mean pixel intensity value separately for each patch. 
x = x - repmat(avg, size(x, 1), 1);

Next, we need to compute . 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

sigma = x * x' / size(x, 2);

(Check the math yourself for correctness.) Here, we assume that x is a data structure that contains one training example per column (so, x is a -by- matrix).

Next, PCA computes the eigenvectors of Σ. One could do this using the Matlab eig function. However, because Σ is a symmetric positive semi-definite matrix, it is more numerically reliable to do this using the svd function. Concretely, if you implement

[U,S,V] = svd(sigma);

then the matrix U will contain the eigenvectors of Sigma (one eigenvector per column, sorted in order from top to bottom eigenvector), and the diagonal entries of the matrix S will contain the corresponding eigenvalues (also sorted in decreasing order). The matrix V will be equal to transpose of U, and can be safely ignored.

(Note: The svd 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.)

Finally, you can compute  and  as follows:

xRot = U' * x;          % rotated version of the data. 
xTilde = U(:,1:k)' * x; % reduced dimension representation of the data, 
                        % where k is the number of eigenvectors to keep

This gives your PCA representation of the data in terms of . Incidentally, if x is a -by- matrix containing all your training data, this is a vectorized implementation, and the expressions above work too for computing x_rot and  for your entire training set all in one go. The resulting x_rot and  will have one column corresponding to each training example.

To compute the PCA whitened data , use

xPCAwhite = diag(1./sqrt(diag(S) + epsilon)) * U' * x;

Since S's diagonal contains the eigenvalues , this turns out to be a compact way of computing  simultaneously for all .

Finally, you can also compute the ZCA whitened data  as:

xZCAwhite = U * diag(1./sqrt(diag(S) + epsilon)) * U' * x;

Exercise:PCA in 2D

PCA, PCA whitening and ZCA whitening in 2D

In this exercise you will implement PCA, PCA whitening and ZCA whitening, as described in the earlier sections of this tutorial, and generate the images shown in the earlier sections yourself. You will build on the starter code that has been provided atpca_2d.zip. You need only write code at the places indicated by "YOUR CODE HERE" in the files. The only file you need to modify ispca_2d.m. Implementing this exercise will make the next exercise significantly easier to understand and complete.

Step 0: Load data

The starter code contains code to load 45 2D data points. When plotted using the scatter function, the results should look like the following:

Step 1: Implement PCA

In this step, you will implement PCA to obtain x_rot, the matrix in which the data is "rotated" to the basis comprising made up of the principal components. As mentioned in the implementation notes, you should make use of MATLAB's svd function here.

Step 1a: Finding the PCA basis

Find  and , and draw two lines in your figure to show the resulting basis on top of the given data points. You may find it useful to use MATLAB's hold on and hold off functions. (After calling hold on, plotting functions such as plot will draw the new data on top of the previously existing figure rather than erasing and replacing it; and hold off turns this off.) You can use plot([x1,x2], [y1,y2], '-') to draw a line between (x1,y1) and (x2,y2). Your figure should look like this:

If you are doing this in Matlab, you will probably get a plot that's identical to ours. However, eigenvectors are defined only up to a sign. I.e., instead of returning  as the first eigenvector, Matlab/Octave could just as easily have returned , and similarly instead of  Matlab/Octave could have returned . So if you wound up with one or both of the eigenvectors pointing in a direction opposite (180 degrees difference) from what's shown above, that's okay too.

Step 1b: Check xRot

Compute xRot, and use the scatter function to check that xRot looks as it should, which should be something like the following:

Because Matlab/Octave could have returned  and/or  instead of  and , it's also possible that you might have gotten a figure which is "flipped" or "reflected" along the - and/or -axis; a flipped/reflected version of this figure is also a completely correct result.

Step 2: Dimension reduce and replot

In the next step, set k, the number of components to retain, to be 1 (we have already done this for you). Compute the resulting xHatand plot the results. You should get the following (this figure should not be flipped along the - or -axis):

Step 3: PCA Whitening

Implement PCA whitening using the formula from the notes. Plot xPCAWhite, and verify that it looks like the following (a figure that is flipped/reflected on either/both axes is also correct):

Step 4: ZCA Whitening

Implement ZCA whitening and plot the results. The results should look like the following (this should not be flipped/reflected along the - or -axis):

Contents

• Step 0: Load data
• Step 1a: Implement PCA to obtain U
• Step 1b: Compute xRot, the projection on to the eigenbasis
• Step 2: Reduce the number of dimensions from 2 to 1.
• Step 3: PCA Whitening
• Step 3: ZCA Whitening
• Congratulations! When you have reached this point, you are done!

close all

%%================================================================

Step 0: Load data

We have provided the code to load data from pcaData.txt into x.
x is a 2 * 45 matrix, where the kth column x(:,k) corresponds to
the kth data point.Here we provide the code to load natural image data into x.
You do not need to change the code below.

x = load('pcaData.txt','-ascii');
figure(1);
scatter(x(1, :), x(2, :));
title('Raw data');


%%================================================================

Step 1a: Implement PCA to obtain U

Implement PCA to obtain the rotation matrix U, which is the eigenbasis
sigma.

% -------------------- YOUR CODE HERE --------------------
u = zeros(size(x, 1)); % You need to compute this
[n m] = size(x);
x = x-repmat(mean(x,2),1,m);%预处理，均值为0
sigma = (1.0/m)*x*x';
[u s v] = svd(sigma);


% --------------------------------------------------------
hold on
plot([0 u(1,1)], [0 u(2,1)]);%画第一条线
plot([0 u(1,2)], [0 u(2,2)]);%第二条线
scatter(x(1, :), x(2, :));
hold off

%%================================================================

Step 1b: Compute xRot, the projection on to the eigenbasis

Now, compute xRot by projecting the data on to the basis defined
by U. Visualize the points by performing a scatter plot.

% -------------------- YOUR CODE HERE --------------------
xRot = zeros(size(x)); % You need to compute this
xRot = u'*x;


% --------------------------------------------------------

% Visualise the covariance matrix. You should see a line across the
% diagonal against a blue background.
figure(2);
scatter(xRot(1, :), xRot(2, :));
title('xRot');

%%================================================================

Step 2: Reduce the number of dimensions from 2 to 1.

Compute xRot again (this time projecting to 1 dimension).
Then, compute xHat by projecting the xRot back onto the original axes
to see the effect of dimension reduction

% -------------------- YOUR CODE HERE --------------------
k = 1; % Use k = 1 and project the data onto the first eigenbasis
xHat = zeros(size(x)); % You need to compute this
xHat = u*([u(:,1),zeros(n,1)]'*x);


% --------------------------------------------------------
figure(3);
scatter(xHat(1, :), xHat(2, :));
title('xHat');


%%================================================================

Step 3: PCA Whitening

Complute xPCAWhite and plot the results.

epsilon = 1e-5;
% -------------------- YOUR CODE HERE --------------------
xPCAWhite = zeros(size(x)); % You need to compute this
xPCAWhite = diag(1./sqrt(diag(s)+epsilon))*u'*x;



% --------------------------------------------------------
figure(4);
scatter(xPCAWhite(1, :), xPCAWhite(2, :));
title('xPCAWhite');

%%================================================================

Step 3: ZCA Whitening

Complute xZCAWhite and plot the results.

% -------------------- YOUR CODE HERE --------------------
xZCAWhite = zeros(size(x)); % You need to compute this
xZCAWhite = u*diag(1./sqrt(diag(s)+epsilon))*u'*x;

% --------------------------------------------------------
figure(5);
scatter(xZCAWhite(1, :), xZCAWhite(2, :));
title('xZCAWhite');

Congratulations! When you have reached this point, you are done!

You can now move onto the next PCA exercise. :)

Exercise:PCA and Whitening
   
   
   
   

    
    
    
    

    
    
    
    
    
    
    
    
PCA and Whitening on natural images
    
    
    
    
In this exercise, you will implement PCA, PCA whitening and ZCA whitening, and apply them to image patches taken from natural images.
    
    
    
    
You will build on the MATLAB starter code which we have provided in pca_exercise.zip. You need only write code at the places indicated by "YOUR CODE HERE" in the files. The only file you need to modify is pca_gen.m.
    
    
    
    
Step 0: Prepare data
    
    
    
    
Step 0a: Load data
    
    
    
    
The starter code contains code to load a set of natural images and sample 12x12 patches from them. The raw patches will look something like this:
    
    
    
    

    
    
    
    
These patches are stored as column vectors  in the  matrix x.
    
    
    
    
Step 0b: Zero mean the data
    
    
    
    
First, for each image patch, compute the mean pixel value and subtract it from that image, this centering the image around zero. You should compute a different mean value for each image patch.
    
    
    
    
Step 1: Implement PCA
    
    
    
    
Step 1a: Implement PCA
    
    
    
    
In this step, you will implement PCA to obtain xrot, the matrix in which the data is "rotated" to the basis comprising the principal components (i.e. the eigenvectors of Σ). Note that in this part of the exercise, you should not whiten the data.
    
    
    
    
Step 1b: Check covariance
    
    
    
    
To verify that your implementation of PCA is correct, you should check the covariance matrix for the rotated data xrot. PCA guarantees that the covariance matrix for the rotated data is a diagonal matrix (a matrix with non-zero entries only along the main diagonal). Implement code to compute the covariance matrix and verify this property. One way to do this is to compute the covariance matrix, and visualise it using the MATLAB command imagesc. The image should show a coloured diagonal line against a blue background. For this dataset, because of the range of the diagonal entries, the diagonal line may not be apparent, so you might get a figure like the one show below, but this trick of visualizing using imagesc will come in handy later in this exercise.
    
    
    
    

    
    
    
    
Step 2: Find number of components to retain
    
    
    
    
Next, choose k, the number of principal components to retain. Pick k to be as small as possible, but so that at least 99% of the variance is retained. In the step after this, you will discard all but the top k principal components, reducing the dimension of the original data to k.
    
    
    
    
Step 3: PCA with dimension reduction
    
    
    
    
Now that you have found k, compute , the reduced-dimension representation of the data. This gives you a representation of each image patch as a k dimensional vector instead of a 144 dimensional vector. If you are training a sparse autoencoder or other algorithm on this reduced-dimensional data, it will run faster than if you were training on the original 144 dimensional data.
    
    
    
    
To see the effect of dimension reduction, go back from  to produce the matrix , the dimension-reduced data but expressed in the original 144 dimensional space of image patches. Visualise  and compare it to the raw data, x. You will observe that there is little loss due to throwing away the principal components that correspond to dimensions with low variation. For comparison, you may also wish to generate and visualise  for when only 90% of the variance is retained.
    
    
    
    
 
     
 
      
 
       
 
       
 
       
 
      
 
      
 
       
Raw images   
 
       
PCA dimension-reduced images (99% variance)
 
       
PCA dimension-reduced images (90% variance)
 
      
 
     
 
    
    
    
    
    
Step 4: PCA with whitening and regularization
    
    
    
    
Step 4a: Implement PCA with whitening and regularization
    
    
    
    
Now implement PCA with whitening and regularization to produce the matrix xPCAWhite. Use the following parameter value:
    
    
    
    
epsilon = 0.1

    
    
    
    
Step 4b: Check covariance
    
    
    
    
Similar to using PCA alone, PCA with whitening also results in processed data that has a diagonal covariance matrix. However, unlike PCA alone, whitening additionally ensures that the diagonal entries are equal to 1, i.e. that the covariance matrix is the identity matrix.
    
    
    
    
That would be the case if you were doing whitening alone with no regularization. However, in this case you are whitening with regularization, to avoid numerical/etc. problems associated with small eigenvalues. As a result of this, some of the diagonal entries of the covariance of your xPCAwhite will be smaller than 1.
    
    
    
    
To verify that your implementation of PCA whitening with and without regularization is correct, you can check these properties. Implement code to compute the covariance matrix and verify this property. (To check the result of PCA without whitening, simply set epsilon to 0, or close to 0, say 1e-10). As earlier, you can visualise the covariance matrix with imagesc. When visualised as an image, for PCA whitening without regularization you should see a red line across the diagonal (corresponding to the one entries) against a blue background (corresponding to the zero entries); for PCA whitening with regularization you should see a red line that slowly turns blue across the diagonal (corresponding to the 1 entries slowly becoming smaller).
    
    
    
    
 
     
 
      
 
       
 
       
 
      
 
      
 
       
 Covariance for PCA whitening with regularization 
 
       
 Covariance for PCA whitening without regularization 
 
      
 
     
 
    
    
    
    
    
Step 5: ZCA whitening
    
    
    
    
Now implement ZCA whitening to produce the matrix xZCAWhite. Visualize xZCAWhite and compare it to the raw data, x. You should observe that whitening results in, among other things, enhanced edges. Try repeating this with epsilon set to 1, 0.1, and 0.01, and see what you obtain. The example shown below (left image) was obtained with epsilon = 0.1.
    
    
    
    
 
     
 
      
 
       
 
       
 
      
 
      
 
       
ZCA whitened images
 
       
Raw images
 
      
 
     
 
    
   
   
   
   
Contents
   
   
   
   

    
    
    
    
 
     
Step 0a: Load data
 
     
Step 0b: Zero-mean the data (by row)
 
     
Step 1a: Implement PCA to obtain xRot
 
     
Step 1b: Check your implementation of PCA
 
     
Step 2: Find k, the number of components to retain
 
     
Step 3: Implement PCA with dimension reduction
 
     
Step 4a: Implement PCA with whitening and regularisation
 
     
Step 4b: Check your implementation of PCA whitening
 
     
Step 5: Implement ZCA whitening
 
    
   
   
   
   
%%================================================================
Step 0a: Load data
Here we provide the code to load natural image data into x.
x will be a 144 * 10000 matrix, where the kth column x(:, k) corresponds to
the raw image data from the kth 12x12 image patch sampled.
You do not need to change the code below.
x = sampleIMAGESRAW();
figure('name','Raw images');
randsel = randi(size(x,2),204,1); % A random selection of samples for visualization
display_network(x(:,randsel));%为什么x有负数还可以显示？

%%================================================================
Step 0b: Zero-mean the data (by row)
You can make use of the mean and repmat/bsxfun functions.
% -------------------- YOUR CODE HERE --------------------
x = x-repmat(mean(x,1),size(x,1),1);%求的是每一列的均值
%x = x-repmat(mean(x,2),1,size(x,2));

%%================================================================
Step 1a: Implement PCA to obtain xRot
Implement PCA to obtain xRot, the matrix in which the data is expressed
with respect to the eigenbasis of sigma, which is the matrix U.
% -------------------- YOUR CODE HERE --------------------
xRot = zeros(size(x)); % You need to compute this
[n m] = size(x);
sigma = (1.0/m)*x*x';
[u s v] = svd(sigma);
xRot = u'*x;


%%================================================================
Step 1b: Check your implementation of PCA
The covariance matrix for the data expressed with respect to the basis U
should be a diagonal matrix with non-zero entries only along the main
diagonal. We will verify this here.
Write code to compute the covariance matrix, covar.
When visualised as an image, you should see a straight line across the
diagonal (non-zero entries) against a blue background (zero entries).
% -------------------- YOUR CODE HERE --------------------
covar = zeros(size(x, 1)); % You need to compute this
covar = (1./m)*xRot*xRot';

% Visualise the covariance matrix. You should see a line across the
% diagonal against a blue background.
figure('name','Visualisation of covariance matrix');
imagesc(covar);

%%================================================================
Step 2: Find k, the number of components to retain
Write code to determine k, the number of components to retain in order
to retain at least 99% of the variance.
% -------------------- YOUR CODE HERE --------------------
k = 0; % Set k accordingly
ss = diag(s);
% for k=1:m
%    if sum(s(1:k))./sum(ss) < 0.99
%        continue;
% end
%其中cumsum(ss)求出的是一个累积向量，也就是说ss向量值的累加值
%并且(cumsum(ss)/sum(ss))<=0.99是一个向量，值为0或者1的向量，为1表示满足那个条件
k = length(ss((cumsum(ss)/sum(ss))<=0.99));

%%================================================================
Step 3: Implement PCA with dimension reduction
Now that you have found k, you can reduce the dimension of the data by
discarding the remaining dimensions. In this way, you can represent the
data in k dimensions instead of the original 144, which will save you
computational time when running learning algorithms on the reduced
representation.
Following the dimension reduction, invert the PCA transformation to produce
the matrix xHat, the dimension-reduced data with respect to the original basis.
Visualise the data and compare it to the raw data. You will observe that
there is little loss due to throwing away the principal components that
correspond to dimensions with low variation.
% -------------------- YOUR CODE HERE --------------------
xHat = zeros(size(x));  % You need to compute this
xHat = u*[u(:,1:k)'*x;zeros(n-k,m)];

% Visualise the data, and compare it to the raw data
% You should observe that the raw and processed data are of comparable quality.
% For comparison, you may wish to generate a PCA reduced image which
% retains only 90% of the variance.

figure('name',['PCA processed images ',sprintf('(%d / %d dimensions)', k, size(x, 1)),'']);
display_network(xHat(:,randsel));
figure('name','Raw images');
display_network(x(:,randsel));

%%================================================================
 
Step 4a: Implement PCA with whitening and regularisation
Implement PCA with whitening and regularisation to produce the matrix
xPCAWhite.
epsilon = 0.1;
xPCAWhite = zeros(size(x));

% -------------------- YOUR CODE HERE --------------------
xPCAWhite = diag(1./sqrt(diag(s)+epsilon))*u'*x;
figure('name','PCA whitened images');
display_network(xPCAWhite(:,randsel));

%%================================================================
Step 4b: Check your implementation of PCA whitening
Check your implementation of PCA whitening with and without regularisation.
PCA whitening without regularisation results a covariance matrix
that is equal to the identity matrix. PCA whitening with regularisation
results in a covariance matrix with diagonal entries starting close to
1 and gradually becoming smaller. We will verify these properties here.
Write code to compute the covariance matrix, covar.
Without regularisation (set epsilon to 0 or close to 0),
when visualised as an image, you should see a red line across the
diagonal (one entries) against a blue background (zero entries).
With regularisation, you should see a red line that slowly turns
blue across the diagonal, corresponding to the one entries slowly
becoming smaller.
% -------------------- YOUR CODE HERE --------------------
covar = (1./m)*xPCAWhite*xPCAWhite';

% Visualise the covariance matrix. You should see a red line across the
% diagonal against a blue background.
figure('name','Visualisation of covariance matrix');
imagesc(covar);

%%================================================================
Step 5: Implement ZCA whitening
Now implement ZCA whitening to produce the matrix xZCAWhite.
Visualise the data and compare it to the raw data. You should observe
that whitening results in, among other things, enhanced edges.
xZCAWhite = zeros(size(x));

% -------------------- YOUR CODE HERE --------------------
xZCAWhite = u*xPCAWhite;

% Visualise the data, and compare it to the raw data.
% You should observe that the whitened images have enhanced edges.
figure('name','ZCA whitened images');
display_network(xZCAWhite(:,randsel));
figure('name','Raw images');
display_network(x(:,randsel));
 

Published with MATLAB® 7.11