Representational dissimilarity metric spaces for stochastic neural networks

Comparing high-dimensional neural response patterns across different animals or artificial deep networks is a fundamental problem in computational neuroscience. There are now many methods for quantifying representational dissimilarity including canonical correlations analysis (CCA), centered kernel alignment (CKA), representational similarity analysis (RSA), shape metrics, and so on. Intuitively, these measures quantify similarity in the geometry of neural responses while removing expected forms of invariance across networks (e.g. permutations over arbitrary neuron indices). However, these methods only compare deterministic representations of external stimuli. Biological networks are essentially never deterministic in this fashion. In fact, the variance of a stimulus-evoked neural response is often larger than its mean, and influential theoretical work has shown that the structure of noise correlations can limit the information capacity of neural circuits. Stochastic responses also arise in the deep learning. literature in many contexts, such as in deep generative modeling or to provide regularization (e.g. dropout). We currently lack rigorous methods to compare representations across these stochastic networks, beyond using their trial-average response (which ignores stochasticity entirely). To fill the gap in the literature, we generalize previously proposed shape metrics to quantify differences in stochastic representations. Stochastic shape metrics are proper metrics – they are non-negative, symmetric, and satisfy 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 deep network attributes (e.g. training hyperparameters) from its position in stochastic (versus deterministic) shape space.