Analysis of the Identifying Regulation With Adversarial Surrogates Algorithm

Given a time-series
$\{\textrm {z}^{k}\}_{\textrm {k=1}}^{\textrm {N}}$
of noisy measured outputs along a single trajectory of a dynamical system, the Identifying Regulation with Adversarial Surrogates (IRAS) algorithm aims to find a non-trivial first integral of the system, that is, a scalar function g such that
$ g\text {(}z^{i}\text {)} \approx g\text {(}z^{j}\text {)}$
, for all i, j. IRAS has been suggested recently and was used successfully in several learning tasks in models from biology and physics. Here, we give the first rigorous analysis of this algorithm in a specific setting. We assume that the observations admit a linear first integral and that they are contaminated by Gaussian noise. We show that in this case the IRAS iterations are closely related to the self-consistent-field (SCF) iterations for solving a generalized Rayleigh quotient minimization problem. Using this approach, we derive several sufficient conditions guaranteeing local convergence of IRAS to the linear first integral.