Universal Approximation of Linear Time-Invariant (LTI) Systems Through RNNs: Power of Randomness in Reservoir Computing

Recurrent neural networks (RNNs) are known to be universal approximators of dynamic systems under fairly mild and general assumptions. However, RNNs usually suffer from the issues of vanishing and exploding gradients in standard RNN training. Reservoir computing (RC), a special RNN where the recurrent weights are randomized and left untrained, has been introduced to overcome these issues and has demonstrated superior empirical performance especially in scenarios where training samples are extremely limited. On the other hand, the theoretical grounding to support this observed performance has not yet been fully developed. In this work, we show that RC can universally approximate a general linear time-invariant (LTI) system. Specifically, we present a clear signal processing interpretation of RC and utilize this understanding in the problem of approximating a generic LTI system. Under this setup, we analytically characterize the optimum probability density function for configuring (instead of training and/or randomly generating) the recurrent weights of the underlying RNN of the RC. Extensive numerical evaluations are provided to validate the optimality of the derived distribution for configuring the recurrent weights of the RC to approximate a general LTI system. Our work results in clear signal processing-based model interpretability of RC and provides theoretical explanation/justification for the power of randomness in randomly generating instead of training RC's recurrent weights. Furthermore, it provides a complete optimum analytical characterization for configuring the untrained recurrent weights, marking an important step towards explainable machine learning (XML) to incorporate domain knowledge for efficient learning.