  A numerical approach is developed for detecting the equivalence of deep
learning architectures. The method is based on generating Mixed Matrix
Ensembles (MMEs) out of deep neural network weight matrices and {\it conjugate
circular ensemble} matching the neural architecture topology. Following this,
the empirical evidence supports the {\it phenomenon} that difference between
spectral densities of neural architectures and corresponding {\it conjugate
circular ensemble} are vanishing with different decay rates at the long
positive tail part of the spectrum i.e., cumulative Circular Spectral
Difference (CSD). This finding can be used in establishing equivalences among
different neural architectures via analysis of fluctuations in CSD. We
investigated this phenomenon for a wide range of deep learning vision
architectures and with circular ensembles originating from statistical quantum
mechanics. Practical implications of the proposed method for artificial and
natural neural architectures discussed such as the possibility of using the
approach in Neural Architecture Search (NAS) and classification of biological
neural networks.
