  Understanding the operation of biological and artificial networks remains a
difficult and important challenge. To identify general principles, researchers
are increasingly interested in surveying large collections of networks that are
trained on, or biologically adapted to, similar tasks. A standardized set of
analysis tools is now needed to identify how network-level covariates -- such
as architecture, anatomical brain region, and model organism -- impact neural
representations (hidden layer activations). Here, we provide a rigorous
foundation for these analyses by defining a broad family of metric spaces that
quantify representational dissimilarity. Using this framework we modify
existing representational similarity measures based on canonical correlation
analysis to satisfy the triangle inequality, formulate a novel metric that
respects the inductive biases in convolutional layers, and identify approximate
Euclidean embeddings that enable network representations to be incorporated
into essentially any off-the-shelf machine learning method. We demonstrate
these methods on large-scale datasets from biology (Allen Institute Brain
Observatory) and deep learning (NAS-Bench-101). In doing so, we identify
relationships between neural representations that are interpretable in terms of
anatomical features and model performance.
