Bounds on Lyapunov Exponents via Entropy Accumulation
Bounds on Lyapunov Exponents via Entropy Accumulation
复制标题
通过熵累加确定李雅普诺夫指数的界限
DOI：
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发表时间：
2019
影响因子：
2.5
通讯作者：
R. Renner
中科院分区：
文献类型：
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作者：
David Sutter;Omar Fawzi;R. Renner
李雅普诺夫指数描述了随机矩阵大积的奇异值的渐近行为。然而，直接计算这些指数往往是不可行的。通过建立Lyapunov指数与信息论工具--熵积累定理之间的联系，我们分别得到了最大和最小Lyapunov指数的上界和下界。这些边界假定随机矩阵独立，是解析的，并且在交换情况下以及在其他情况下都是紧的。它们可以用只涉及单个矩阵而不涉及大乘积的优化问题来表示。利用凸优化理论可以有效地求出最大Lyapunov指数的上界。
Lyapunov exponents describe the asymptotic behavior of the singular values of large products of random matrices. A direct computation of these exponents is however often infeasible. By establishing a link between Lyapunov exponents and an information theoretic tool called entropy accumulation theorem we derive an upper and a lower bound for the maximal and minimal Lyapunov exponent, respectively. The bounds assume independence of the random matrices, are analytical, and are tight in the commutative case as well as in other scenarios. They can be expressed in terms of an optimization problem that only involves single matrices rather than large products. The upper bound for the maximal Lyapunov exponent can be evaluated efficiently via the theory of convex optimization.