  Batch Normalization (BN) is essential to effectively train state-of-the-art
deep Convolutional Neural Networks (CNN). It normalizes inputs to the layers
during training using the statistics of each mini-batch. In this work, we study
BN from the viewpoint of Fisher kernels. We show that assuming samples within a
mini-batch are from the same probability density function, then BN is identical
to the Fisher vector of a Gaussian distribution. That means BN can be explained
in terms of kernels that naturally emerge from the probability density function
of the underlying data distribution. However, given the rectifying
non-linearities employed in CNN architectures, distribution of inputs to the
layers show heavy tail and asymmetric characteristics. Therefore, we propose
approximating underlying data distribution not with one, but a mixture of
Gaussian densities. Deriving Fisher vector for a Gaussian Mixture Model (GMM),
reveals that BN can be improved by independently normalizing with respect to
the statistics of disentangled sub-populations. We refer to our proposed soft
piecewise version of BN as Mixture Normalization (MN). Through extensive set of
experiments on CIFAR-10 and CIFAR-100, we show that MN not only effectively
accelerates training image classification and Generative Adversarial networks,
but also reaches higher quality models.
