  Despite substantial efforts, neural network interpretability remains an
elusive goal, with previous research failing to provide succinct explanations
of most single neurons' impact on the network output. This limitation is due to
the polysemantic nature of most neurons, whereby a given neuron is involved in
multiple unrelated network states, complicating the interpretation of that
neuron. In this paper, we apply tools developed in neuroscience and information
theory to propose both a novel practical approach to network interpretability
and theoretical insights into polysemanticity and the density of codes. We
infer levels of redundancy in the network's code by inspecting the
eigenspectrum of the activation's covariance matrix. Furthermore, we show how
random projections can reveal whether a network exhibits a smooth or
non-differentiable code and hence how interpretable the code is. This same
framework explains the advantages of polysemantic neurons to learning
performance and explains trends found in recent results by Elhage et
al.~(2022). Our approach advances the pursuit of interpretability in neural
networks, providing insights into their underlying structure and suggesting new
avenues for circuit-level interpretability.
