Memory of recurrent networks: Do we compute it right?

TL;DR

This paper investigates the discrepancy between theoretical and empirical memory capacity estimates in linear echo state networks, identifying numerical issues and proposing robust methods that align empirical results with theoretical bounds.

Contribution

It reveals numerical causes of inaccurate memory capacity estimations and introduces robust approaches that ensure empirical results match theoretical predictions in linear recurrent networks.

Findings

Numerical issues cause discrepancies in memory capacity estimates.

Ignoring Krylov structure introduces gaps between theory and empirical results.

Proposed methods recover memory curves that align with theoretical bounds.

Abstract

Numerical evaluations of the memory capacity (MC) of recurrent neural networks reported in the literature often contradict well-established theoretical bounds. In this paper, we study the case of linear echo state networks, for which the total memory capacity has been proven to be equal to the rank of the corresponding Kalman controllability matrix. We shed light on various reasons for the inaccurate numerical estimations of the memory, and we show that these issues, often overlooked in the recent literature, are of an exclusively numerical nature. More explicitly, we prove that when the Krylov structure of the linear MC is ignored, a gap between the theoretical MC and its empirical counterpart is introduced. As a solution, we develop robust numerical approaches by exploiting a result of MC neutrality with respect to the input mask matrix. Simulations show that the memory curves that are recovered using the proposed methods fully agree with the theory.