Similarity of Neural Network Representations Revisited

Recent work has sought to understand the behavior of neural networks by comparing representations between layers and between different trained models. We examine methods for comparing neural network representations based on canonical correlation analysis (CCA). We show that CCA belongs to a family of statistics for measuring multivariate similarity, but that neither CCA nor any other statistic that is invariant to invertible linear transformation can measure meaningful similarities between representations of higher dimension than the number of available data points. We introduce a similarity index that measures the relationship between representational similarity matrices and does not suffer from this limitation. We show that this similarity index is equivalent to centered kernel-tangent alignment (KTA) and is also closely connected to CCA. Unlike other methods, KTA can reliably identify correspondences between representations of layers in networks trained from different initializations.