Reconstructing Input Sequences from Reservoir States: Readouts that Compensate the Transfer Characteristic of the Nodes and the Limits of Information Recovery

Reservoir computing uses the nonlinearity of its elements to generate features and reads out the state through a linear map.However, the same nonlinearity acts as the transfer characteristic of each element, compressing and saturating the input;in physical implementations, noise and finite-precision readout further compound this effect. How much information that a linear map cannot recover remains in the state has not been measured. This paper takes echo state networks as the subject and adopts the task of reconstructing the input sequence from the state sequence as the evaluation axis, proposing two readouts that compensate for the transfer characteristic: a per-timestep multilayer perceptron and a GRU regressor with temporal context. On the intermediate layers of a pretrained language model, the former reduces the error of the linear readout by more than a factor of~$40$, and the latter by more than a factor of~$70$. When the input amplitude is increased, the linear readout error worsens by a factor of~$140$, whereas the former stays within a factor of~$14$ and the latter within a factor of~$3$; the compensation is insensitive to the operating point. A generative readout trained with the same backbone as the latter yields errors only $3$ to $7$ times those of the latter. With the GRU regressor fixed, an $8$-way split at constant element count halves the error, and an error-correcting code on the input side achieves error-free recovery at a coded bit error rate of~$10^{-3}$, at the cost of rate.