Computationally Efficient Approximations of the Joint Spectral Radius

• Authors: Vincent D. Blondel;Yurii Nesterov
• Venue: SIAM Journal on Matrix Analysis and Applications
• Year: 2005

The joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate that can be obtained by forming long products of matrices taken from the set. This quantity appears in a number of application contexts but is notoriously difficult to compute and to approximate. We introduce in this paper a procedure for approximating the joint spectral radius of a finite set of matrices with arbitrary high accuracy. Our approximation procedure is polynomial in the size of the matrices once the number of matrices and the desired accuracy are fixed. For the special case of matrices with nonnegative entries we give elementary proofs of simple inequalities that we then use to obtain approximations of arbitrary high accuracy. From these inequalities it follows that the spectral radius of matrices with nonnegative entries is given by the simple expression $$ \rho(A_1, \ldots, A_m)= \lim_{k \rightarrow \infty} \rho^{1/k} ( A_1^{\otimes k} + \cdots + A_m^{\otimes k}), $$ where it is somewhat surprising to notice that the right-hand side does not directly involve any mixed product between the matrices. ($A^{\otimes k}$ denotes the $k$th Kronecker power of $A$.)For matrices with arbitrary entries (not necessarily nonnegative), we introduce an approximation procedure based on semidefinite liftings that can be implemented in a recursive way. For two matrices, even the first step of the procedure gives an approximation whose relative accuracy is at least ${1 / \sqrt{2}}$, that is, more than $70\%$. The subsequent steps improve the accuracy but also increase the dimension of the auxiliary problems from which the approximation can be found.Our approximation procedures provide approximations of relative accuracy $1-\epsilon$ in time polynomial in $n^{(\ln m) /\epsilon}$, where $m$ is the number of matrices and $n$ is their size. These bounds are close from optimality since we show that, unless P=NP, no approximation algorithm is possible that provides a relative accuracy of $1-\epsilon$ and runs in time polynomial in $n$ and $1/\epsilon$.As a by-product of our results we prove that a widely used approximation of the joint spectral radius based on common quadratic Lyapunov functions (or on ellipsoid norms) has relative accuracy $1/\sqrt m$, where $m$ is the number of matrices.