On the Number of Linear Regions of Convolutional Neural Networks

One fundamental problem in deep learning is understanding the outstanding performance of deep Neural Networks (NNs) in practice. One explanation for the superiority of NNs is that they can realize a large class of complicated functions, i.e., they have powerful expressivity. The expressivity of a ReLU NN can be quantified by the maximal number of linear regions it can separate its input space into. In this paper, we provide several mathematical results needed for studying the linear regions of CNNs, and use them to derive the maximal and average numbers of linear regions for one-layer ReLU CNNs. Furthermore, we obtain upper and lower bounds for the number of linear regions of multi-layer ReLU CNNs. Our results suggest that deeper CNNs have more powerful expressivity than their shallow counterparts, while CNNs have more expressivity than fully-connected NNs per parameter.

Multi-Granularity Canonical Appearance Pooling for Remote Sensing Scene Classification

Recognising remote sensing scene images remains challenging due to large visual-semantic discrepancies. These mainly arise due to the lack of detailed annotations that can be employed to align pixel-level representations with high-level semantic labels. As the tagging process is labour-intensive and subjective, we hereby propose a novel Multi-Granularity Canonical Appearance Pooling (MG-CAP) to automatically capture the latent ontological structure of remote sensing datasets. We design a granular framework that allows progressively cropping the input image to learn multi-grained features. For each specific granularity, we discover the canonical appearance from a set of pre-defined transformations and learn the corresponding CNN features through a maxout-based Siamese style architecture. Then, we replace the standard CNN features with Gaussian covariance matrices and adopt the proper matrix normalisations for improving the discriminative power of features. Besides, we provide a stable solution for training the eigenvalue-decomposition function (EIG) in a GPU and demonstrate the corresponding back-propagation using matrix calculus. Extensive experiments have shown that our framework can achieve promising results in public remote sensing scene datasets.