PCA

http://deeplearning.stanford.edu/wiki/index.php/PCA

Principal Components Analysis (PCA) is a dimensionality reduction algorithm that can be used to significantly speed up your unsupervised feature learning algorithm.

example

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.

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:

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.)

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.

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

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

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

Number of components to retain

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.

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.

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.

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

, for all