On the Algebraic Structure of Certain Partially Observable Finite-State Markov Processes

We consider a class of nonlinear estimation problems possessing certain algebraic properties, and we exploit these properties in order to study the computational complexity of nonlinear estimation algorithms. Specifically, we define a class of finite-state Markov processes evolving on finite groups and consider noisy observations of these processes. By introducing some concepts from the theory of representations of finite groups, we are able to define a pair of "dual" filtering algorithms. We then study several specific classes of groups in detail, and, by developing a generalization of the fast Fourier transform algorithm, we derive an efficient nonlinear filtering algorithm. A continuous-time version of these ideas is developed for cyclic groups.