Multiply-rooted multiscale models for large-scale estimation

Divide-and-conquer or multiscale techniques have become popular for solving large statistical estimation problems. The methods rely on defining a state which conditionally decorrelates the large problem into multiple subproblems, each more straightforward than the original. However this step cannot be carried out for asymptotically large problems since the dimension of the state grows without bound, leading to problems of computational complexity and numerical stability. In this paper, we propose a new approach to hierarchical estimation in which the conditional decorrelation of arbitrarily large regions is avoided, and the problem is instead addressed piece-by-piece. The approach possesses promising attributes: it is not a local method—the estimate at every point is based on all measurements; it is numerically stable for problems of arbitrary size; and the approach retains the benefits of the multiscale framework on which it is based: a broad class of statistical models, a stochastic realization theory, an algorithm to calculate statistical likelihoods, and the ability to fuse local and nonlocal measurements.