Brief technical note on linearizing recurrent neural networks (RNNs) before vs after the pointwise nonlinearity

Linearization of the dynamics of recurrent neural networks (RNNs) is often used to study their properties. The same RNN dynamics can be written in terms of the activations" (the net inputs to each unit, before its pointwise nonlinearity) or in terms of the activities" (the output of each unit, after its pointwise nonlinearity); the two corresponding linearizations are different from each other. This brief and informal technical note describes the relationship between the two linearizations, between the left and right eigenvectors of their dynamics matrices, and shows that some context-dependent effects are readily apparent under linearization of activity dynamics but not linearization of activation dynamics.