Fast algorithms for approximate stochastic p-th roots of large transition matrices

Develop a fast-converging algorithm to compute, for a given stochastic transition matrix M and integer p > 1, a stochastic matrix H that approximates a stochastic p-th root of M by minimizing a matrix divergence K(M, H^p) (for example, Kullback–Leibler divergence or Frobenius norm), with scalability to matrices of the sizes encountered in large origin–destination mobility networks.

Background

To mitigate mobility-aliasing arising from coarse temporal aggregation, the paper proposes reconstructing finer time-scale dynamics by inserting p shorter-step transition matrices whose product approximates the observed coarse-grained transition matrix. Exact stochastic p-th roots may not exist, so the authors formulate an optimization to find an approximate root by minimizing a divergence K(M, B^p) over stochastic matrices B.

They note that for the large transition matrices arising in their application (e.g., GH5 geohash OD networks at 3-hour resolution), they could not find fast-converging algorithms to compute such approximate roots, identifying a pressing algorithmic gap for scalable solutions.