A topological approach to exploring convolutional neural networks

Motivated by the elusive understanding concerning convolution neural networks (CNNs) in view of topology, we present two theoretical frameworks to interpret two topics by using topological data analysis. The first one reveals the topological essence of CNN filters. Our theory first abstracts a topological representation of how the features locate for a CNN filter, named feature topology, and characterises it by defining the starting edge density. We reveal a principle of CNN filters: tending to organize the feature topologies for the same category, and thus propose the SED Distribution to statistically describe such an organization. We demonstrate the effectiveness of CNN filters reflects in the compactness of SED Distribution, and introduce filter entropy to measure it. Remarkably, the variation of filter entropy during training reveals the essence of CNN training: a filter-entropy-decrease process. Also, based on the principle, we give a metric to assess the filter performance. The second one investigates the inter-class distinguishability in a model-agnostic way. For each class, we propose the MBC Distribution, a distribution that could differentiate categories by characterising the intrinsic organization of the given category. As for multi-classes, we introduce the category distance which metricizes the distance between two categories, and moreover propose the CD Matrix that comprehensively evaluates not just the distinguishability between each two category pair but the distinguishable degree for each category. Finally, our experiment results confirm our theories.Comment: 8 pages, 4 figures, pnas manuscrip