Establishing associations between the structure and the generalisation ability of deep neural networks (DNNs) is a challenging task in modern machine learning. Producing solutions to this challenge will bring progress both in the theoretical understanding of DNNs and in building new architectures efficiently. In this work, we address this challenge by developing a new complexity measure based on the concept of {Periodic Spectral Ergodicity} (PSE) originating from quantum statistical mechanics. Based on this measure a technique is devised to quantify the complexity of deep neural networks from the learned weights and traversing the network connectivity in a sequential manner, hence the term cascading PSE (cPSE), as an empirical complexity measure. This measure will capture both topological and internal neural processing complexity simultaneously. Because of this cascading approach, i.e., a symmetric divergence of PSE on the consecutive layers, it is possible to use this measure for Neural Architecture Search (NAS). We demonstrate the usefulness of this measure in practice on two sets of vision models, ResNet and VGG, and sketch the computation of cPSE for more complex network structures.
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