Power Normalizations (PN) are very useful non-linear operators in the context of Bag-of-Words data representations as they tackle problems such as feature imbalance. In this paper, we reconsider these operators in the deep learning setup by introducing a novel layer that implements PN for non-linear pooling of feature maps. Specifically, by using a kernel formulation, our layer combines the feature vectors and their respective spatial locations in the feature maps produced by the last convolutional layer of CNN. Linearization of such a kernel results in a positive definite matrix capturing the second-order statistics of the feature vectors, to which PN operators are applied. We study two types of PN functions, namely (i) MaxExp and (ii) Gamma, addressing their role and meaning in the context of nonlinear pooling. We also provide a probabilistic interpretation of these operators and derive their surrogates with well-behaved gradients for end-to-end CNN learning. We apply our theory to practice by implementing the PN layer on a ResNet-50 model and showcase experiments on four benchmarks for fine-grained recognition, scene recognition, and material classification. Our results demonstrate state-of-the-art performance across all these tasks.

本文研究了在深度学习中引入Power Normalizations（PN）操作作为一种新的layer，实现对feature map的非线性池化，以解决feature不平衡问题，通过对PN函数的探索和实践，将其与ResNet-50模型相结合，应用于细粒度识别等四个benchmark中，获得了最先进的性能。

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