ZCA whitening in python for Machine learning

Question

I am training 1000 images of 28x28 size. But before training, I am performing ZCA whitening on my data by taking the reference from How to implement ZCA Whitening? Python .

Since I have 1000 data images of size 28x28, after flattening, it becomes 1000x784. But as given in the code below, whether X is my image dataset of 1000x784?

If it is so, then it means the ZCAMatrix size is 1000x1000. In this case, for prediction I have a image of size 28x28, or we can say, size of 1x784.So it doesn't make sense to multiply ZCAMatrix to the image.

So I think, X is the transpose of image data set. Am I right? If I am right, then the size of ZCAMatrix is 784x784.

Now how should I calculate the ZCA whitened image, whether I should use np.dot(ZCAMatrix, transpose_of_image_to_be_predict) or np.dot(image_to_be_predict, ZCAMatrix) ? Suggestion would be greatly appreciate.

def zca_whitening_matrix(X):
    """
    Function to compute ZCA whitening matrix (aka Mahalanobis whitening).
    INPUT:  X: [M x N] matrix.
        Rows: Variables
        Columns: Observations
    OUTPUT: ZCAMatrix: [M x M] matrix
    """
    # Covariance matrix [column-wise variables]: Sigma = (X-mu)' * (X-mu) / N
    sigma = np.cov(X, rowvar=True) # [M x M]
    # Singular Value Decomposition. X = U * np.diag(S) * V
    U,S,V = np.linalg.svd(sigma)
        # U: [M x M] eigenvectors of sigma.
        # S: [M x 1] eigenvalues of sigma.
        # V: [M x M] transpose of U
    # Whitening constant: prevents division by zero
    epsilon = 1e-5
    # ZCA Whitening matrix: U * Lambda * U'
    ZCAMatrix = np.dot(U, np.dot(np.diag(1.0/np.sqrt(S + epsilon)), U.T)) # [M x M]
    return ZCAMatrix

And an example of the usage:

X = np.array([[0, 2, 2], [1, 1, 0], [2, 0, 1], [1, 3, 5], [10, 10, 10] ]) # Input: X [5 x 3] matrix
ZCAMatrix = zca_whitening_matrix(X) # get ZCAMatrix
ZCAMatrix # [5 x 5] matrix
xZCAMatrix = np.dot(ZCAMatrix, X) # project X onto the ZCAMatrix
xZCAMatrix # [5 x 3] matrix

Answer 1

I got the reference from the Keras code available here .

It is very clear that in my case the co-variance matrix will give 784x784 matrix, on which Singular Value Decomposition is performed. It gives 3 matrix that is used to calculate the principal_components , and that principal_components is used to find the ZCA whitened data.

Now my question was

how should I calculate the ZCA whitened image, whether I should use np.dot(ZCAMatrix, transpose_of_image_to_be_predict) or np.dot(image_to_be_predict, ZCAMatrix)? Suggestion would be greatly appreciate.

For this I got the reference from here .

Here I need to use np.dot(image_to_be_predict, ZCAMatrix) to calculate the ZCA whitened image.