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.

翻译：理解神经活动的统计与几何属性如何影响性能，是理论神经科学和深度学习中的关键问题。本文计算了物体表征之间的相关性如何影响容量（一种线性可分性度量）。我们证明：对于球状物体流形，引入质心间相关性实际上会拉近球面间距，而引入轴向间相关性则会有效缩小其半径，从而揭示了相关性与几何结构在分类问题上的对偶性。随后，我们将研究结果应用于准确估计深度网络数据的容量。