Theories and Ultra Efficient Computation of Joint Spectral Radius for Estimating First Passage Time Distribution of Markov Set-Chain

Abstract

This technical note is concerned with the tail distribution of the first passage time of Markov set chains (MSC). An original two-part idea-a more progressive relation and a sortedness test-is conceived to characterize such chains. The theoretical construction based on this idea further results in an algorithm that can compute the tightest exponent bound of the tail distribution for high-dimensional problem instances with surprising ease. To understand the computational implication of this algorithm, note that the problem is equivalent to computing the joint spectral radius (JSR) of a special independent column polytope (one that defines Markov set chains) of nonnegative matrices. In this context, the reported algorithm can compute the exact JSR value for cases of 100 × 100 matrices in less than a second in Matlab. Problems of this size is far beyond the scope of known JSR techniques. It is worth noting that the fields of MSC and JSR have not had significant overlap as one may expect, despite their conceptual akiness. Meanwhile, the present technical note is a contribution that belongs to both fields.