Analysis of the Identifying Regulation with Adversarial Surrogates Algorithm
Ron Teichner, Ron Meir, Michael Margaliot
TL;DR
This work provides the first rigorous analysis of the Identifying Regulation with Adversarial Surrogates (IRAS) algorithm in a setting where observations admit a linear first integral under Gaussian noise. It reveals that each IRAS iteration solves a generalized Rayleigh-quotient problem, i.e., a self-consistent-field (SCF) step, and identifies conditions under which the linear invariant is a stable equilibrium of IRAS. The analysis yields explicit convergence criteria (involving the surrogate noise level $\bar{\sigma}$ relative to the data noise $\sigma$) and demonstrates local convergence to the true invariant, with supporting examples including a 3D case and a biological RFMR application. The work connects IRAS to a broader class of Rayleigh-quotient-based methods and provides a foundation for extending IRAS to nonlinear invariants and improved algorithms with theoretical guarantees, enhancing data-driven discovery of conserved quantities in complex systems.
Abstract
Given a time-series of noisy measured outputs of a dynamical system z[k], k=1...N, the Identifying Regulation with Adversarial Surrogates (IRAS) algorithm aims to find a non-trivial first integral of the system, namely, a scalar function g() such that g(z[i]) = g(z[j]), 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 correct first integral.