Lyapunov Exponent of Rank One Matrices: Ergodic Formula and Inapproximability of the Optimal Distribution
Jason M. Altschuler, Pablo A. Parrilo · 2019
The Lyapunov exponent corresponding to a set of square matrices A = {A1, ... , An} and a probability distribution p over {1, ... , n} is λ(A, p) := limk→∞1/k E log ||Aσk⋯ Aσ2Aσ1||, where σiare i.i.d. according to p. This quantity is of fundamental importance to control theory since it determines the asymptotic convergence rate eλ(A,p)of the stochastic linear dynamical system xk+1= Aσkxk. This paper investigates the following “design problem”: given A, compute the distribution p minimizing λ(A, p). Our main result is that it is NP-hard to decide whether there exists a distribution p for which λ(A, p) <; 0, i.e. it is NP-hard to decide whether this dynamical system can be stabilized. This hardness result holds even in the “simple” case where A contains only rank-one matrices. Somewhat surprisingly, this is in stark contrast to the Joint Spectral Radius - the deterministic kindred of the Lyapunov exponent - for which the analogous optimization problem over switching rules is known to be exactly computable in polynomial time for rank-one matrices. To prove this hardness result, we first observe that the Lyapunov exponent of rank-one matrices admits a simple formula and in fact is a quadratic form in p. Hardness of the design problem is shown through a reduction from the Independent Set problem. Along the way, simple examples are given illustrating that p → λ(A, p) is neither convex nor concave in general, and a connection is made to the fact that the Martin distance on the (1, n) Grassmannian is not a metric.