  Understanding how the statistical and geometric properties of neural activity
relate to performance is a key problem in theoretical neuroscience and deep
learning. Here, we calculate how correlations between object representations
affect the capacity, a measure of linear separability. We show that for
spherical object manifolds, introducing correlations between centroids
effectively pushes the spheres closer together, while introducing correlations
between the axes effectively shrinks their radii, revealing a duality between
correlations and geometry with respect to the problem of classification. We
then apply our results to accurately estimate the capacity of deep network
data.
