Sparse and Constrained State Estimation: A Review of Bayesian Recursion, and Variable-Splitting-Type Methods

This review surveys the mathematical theory and methods of state estimation for stochastic dynamical systems when the Gaussian-linear-least-squares paradigm fails, focusing on the role of sparsity-promoting and constraint-encoding regularization. The exposition is organized around a single regularized estimation functional in which the weighted least-squares terms of classical filtering and smoothing are augmented by nonsmooth penalty functions and indicator functions of constraint sets. Bayesian recursion, Gaussian approximation, sequential Monte Carlo, and moving-horizon optimization emerge as different structural assumptions on this common objective. Recent work is reviewed through five relaxations of the classical assumptions: heavy-tailed and non-Gaussian noise, unknown or misspecified models, distributed and networked architectures, unused structural prior information such as sparsity and safety constraints, and switching or unknown external inputs. Emphasis is placed on variable-splitting and augmented-Lagrangian techniques that interface Gaussian recursions with nonsmooth optimization, on convergence and stability guarantees, and on the mathematical challenges that remain open, including convergence rates for alternating minimization, statistical efficiency relative to least-squares baselines, and provable guarantees for learning-enhanced filters.