Deceiving the CKA Similarity Measure in Deep Learning

Understanding the behaviour of trained deep neural networks is a critical step in allowing reliable deployment of these networks in critical applications. One direction for obtaining insights on neural networks is through comparison of their internal representations. Comparing neural representations in neural networks is thus a challenging but important problem, which has been approached in different ways. The Centered Kernel Alignment (CKA) similarity metric, particularly its linear variant, has recently become a popular approach and has been widely used to compare representations of a network's different layers, of architecturally similar networks trained differently, or of models with different architectures trained on the same data. A wide variety of conclusions about similarity and dissimilarity of these various representations have been made using CKA. In this work we present an analysis that formally characterizes CKA sensitivity to a large class of simple transformations, which can naturally occur in the context of modern machine learning. This provides a concrete explanation of CKA sensitivity to outliers and to transformations that preserve the linear separability of the data, an important generalization attribute. Finally we propose an optimization-based approach for modifying representations to maintain functional behaviour while changing the CKA value. Our results illustrate that, in many cases, the CKA value can be easily manipulated without substantial changes to the functional behaviour of the models, and call for caution when leveraging activation alignment metrics.