Lyapunov Exponent of Rank One Matrices: Ergodic Formula and Inapproximability of the Optimal Distribution

Abstract: The Lyapunov exponent corresponding to a set of square matrices $\mathcal{A} = {A_1, \dots, A_n }$ and a probability distribution $p$ over ${1, \dots, n}$ is $\lambda(\mathcal{A},p) := \lim_{k \to \infty} \frac{1}{k} \,\mathbb{E} \log |A_{\sigma_k} \cdots A_{\sigma_2}A_{\sigma_1}|$, where $\sigma_i$ are i.i.d. according to $p$. This quantity is of fundamental importance to control theory since it determines the asymptotic convergence rate $e^{\lambda(\mathcal{A},p)}$ of the stochastic linear dynamical system $x_{k+1} = A_{\sigma_k} x_k$. This paper investigates the following "design problem": given $\mathcal{A}$, compute the distribution $p$ minimizing $\lambda(\mathcal{A},p)$. Our main result is that it is NP-hard to decide whether there exists a distribution $p$ for which $\lambda(\mathcal{A},p)< 0$, 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 $\mathcal{A}$ 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 for rank-one matrices is known to be exactly computable in polynomial time. To prove this hardness result, we first observe via Birkhoff's Ergodic Theorem 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 $p \mapsto \lambda(\mathcal{A},p)$ is neither convex nor concave in general. We conclude with extensions to continuous distributions, exchangeable processes, Markov processes, and stationary ergodic processes.