Batch Normalization (BN) techniques have been proposed to reduce the so-called Internal Covariate Shift (ICS) by attempting to keep the distributions of layer outputs unchanged. Experiments have shown their effectiveness on training deep neural networks. However, since only the first two moments are controlled in these BN techniques, it seems that a weak constraint is imposed on layer distributions and furthermore whether such constraint can reduce ICS is unknown. Thus this paper proposes a measure for ICS by using the Earth Mover (EM) distance and then derives the upper and lower bounds for the measure to provide a theoretical analysis of BN. The upper bound has shown that BN techniques can control ICS only for the outputs with low dimensions and small noise whereas their control is NOT effective in other cases. This paper also proves that such control is just a bounding of ICS rather than a reduction of ICS. Meanwhile, the analysis shows that the high-order moments and noise, which BN cannot control, have great impact on the lower bound. Based on such analysis, this paper furthermore proposes an algorithm that unitizes the outputs with an adjustable parameter to further bound ICS in order to cope with the problems of BN. The upper bound for the proposed unitization is noise-free and only dominated by the parameter. Thus, the parameter can be trained to tune the bound and further to control ICS. Besides, the unitization is embedded into the framework of BN to reduce the information loss. The experiments show that this proposed algorithm outperforms existing BN techniques on CIFAR-10, CIFAR-100 and ImageNet datasets.
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