  Quantifying similarity between neural representations -- e.g. hidden layer
activation vectors -- is a perennial problem in deep learning and neuroscience
research. Existing methods compare deterministic responses (e.g. artificial
networks that lack stochastic layers) or averaged responses (e.g.,
trial-averaged firing rates in biological data). However, these measures of
_deterministic_ representational similarity ignore the scale and geometric
structure of noise, both of which play important roles in neural computation.
To rectify this, we generalize previously proposed shape metrics (Williams et
al. 2021) to quantify differences in _stochastic_ representations. These new
distances satisfy the triangle inequality, and thus can be used as a rigorous
basis for many supervised and unsupervised analyses. Leveraging this novel
framework, we find that the stochastic geometries of neurobiological
representations of oriented visual gratings and naturalistic scenes
respectively resemble untrained and trained deep network representations.
Further, we are able to more accurately predict certain network attributes
(e.g. training hyperparameters) from its position in stochastic (versus
deterministic) shape space.
