Improvements to numerical stability and performance for ZCA whitening (…

…#9025)

* Implemented zca_whitening without forming covariance matrix or explicit call to diag for eigenvalues

* Made principal_components self.principal_components

* Fixed mistake with zca_epsilon

* Style fixes

keras/preprocessing/image.py

@@ -748,9 +748,10 @@ def fit(self, x,
 
         if self.zca_whitening:
             flat_x = np.reshape(x, (x.shape[0], x.shape[1] * x.shape[2] * x.shape[3]))
-            sigma = np.dot(flat_x.T, flat_x) / flat_x.shape[0]
-            u, s, _ = linalg.svd(sigma)
-            self.principal_components = np.dot(np.dot(u, np.diag(1. / np.sqrt(s + self.zca_epsilon))), u.T)
+            num_examples = flat_x.shape[0]
+            u, s, vt = linalg.svd(flat_x / np.sqrt(num_examples))
+            s_expand = np.hstack((s, np.zeros(vt.shape[0] - num_examples, dtype=flat_x.dtype)))
+            self.principal_components = (vt.T / np.sqrt(s_expand ** 2 + self.zca_epsilon)).dot(vt)
 
 
 class Iterator(Sequence):