The quantitative analysis of information structure through a deep neural network (DNN) can unveil new insights into the theoretical performance of DNN architectures. Two very promising avenues of research towards quantitative information structure analysis are: 1) layer similarity (LS) strategies focused on the inter-layer feature similarity, and 2) intrinsic dimensionality (ID) strategies focused on layer-wise data dimensionality using pairwise information. Inspired by both LS and ID strategies for quantitative information structure analysis, we introduce two novel complimentary methods for inter-layer information similarity assessment premised on the interesting idea of studying a data sample's neighbourhood dynamics as it traverses through a DNN. More specifically, we introduce the concept of Nearest Neighbour Topological Similarity (NNTS) for quantifying the information topology similarity between layers of a DNN. Furthermore, we introduce the concept of Nearest Neighbour Topological Persistence (NNTP) for quantifying the inter-layer persistence of data neighbourhood relationships throughout a DNN. The proposed strategies facilitate the efficient inter-layer information similarity assessment by leveraging only local topological information, and we demonstrate their efficacy in this study by performing analysis on a deep convolutional neural network architecture on image data to study the insights that can be gained with respect to the theoretical performance of a DNN.
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