Efficient Training of Minimal and Maximal Low-Rank Recurrent Neural Networks

Low-rank recurrent neural networks (RNNs) provide a powerful framework for characterizing how neural systems solve complex cognitive tasks. However, fitting and interpreting these networks remains an important open problem. In this paper, we develop new methods for efficiently fitting low-rank RNNs in ''teacher-training'' settings. In particular, we build upon the neural engineering framework (NEF), in which RNNs are viewed as approximating an ordinary differential equation (ODE) of interest using a set of random nonlinear basis functions. This view provides geometric insight into how the choice of neural nonlinearity (e.g. tanh, ReLU) and the distribution of model parameters affects an RNN's representational capacity. We show that this perspective leads to an online training method that achieves higher accuracy with smaller networks than previous methods such as FORCE, and outperform backprop-trained networks of similar size while requiring substantially less training time. We then consider the problem of finding minimal and maximal low-RNNs for approximating a target dynamical system. We show that a variant of orthogonal matching pursuit (OMP) can be used to find the smallest RNN for a dynamical system of interest. At the other extreme, a dual space formulation allows for efficient fitting of infinite low-rank RNNs, which provide a Gaussian Process (GP) prior over dynamical systems. We use the resulting GP marginal likelihood to optimize the hyperparameters governing neural activation functions, which leads to improved training performance even for finite RNNs. Finally, we describe active learning methods for low-rank RNNs, which speed up training through the selection of maximally informative activity patterns.