  Understanding the deep representations of complex networks is an important
step of building interpretable and trustworthy machine learning applications in
the age of internet. Global surrogate models that approximate the predictions
of a black box model (e.g. an artificial or biological neural net) are usually
used to provide valuable theoretical insights for the model interpretability.
In order to evaluate how well a surrogate model can account for the
representation in another model, we need to develop inference methods for model
comparison. Previous studies have compared models and brains in terms of their
representational geometries (characterized by the matrix of distances between
representations of the input patterns in a model layer or cortical area). In
this study, we propose to explore these summary statistical descriptions of
representations in models and brains as part of a broader class of statistics
that emphasize the topology as well as the geometry of representations. The
topological summary statistics build on topological data analysis (TDA) and
other graph-based methods. We evaluate these statistics in terms of the
sensitivity and specificity that they afford when used for model selection,
with the goal to relate different neural network models to each other and to
make inferences about the computational mechanism that might best account for a
black box representation. These new methods enable brain and computer
scientists to visualize the dynamic representational transformations learned by
brains and models, and to perform model-comparative statistical inference.
