A Computationally Efficient Maximum A Posteriori Sequence Estimation via Stein Variational Inference

State estimation in robotic systems presents significant challenges, particularly due to the prevalence of multimodal posterior distributions in real-world scenarios. One effective strategy for handling such complexity is to compute maximum a posteriori (MAP) sequences over a discretized or sampled state space, which enables a concise representation of the most likely state trajectory. However, this approach often incurs substantial computational costs, especially in high-dimensional settings. In this article, we propose a novel MAP sequence estimation method, \textsf{Stein-MAP-Seq}, which effectively addresses multimodality while substantially reducing computational and memory overhead. Our key contribution is a sequential variational inference framework that captures temporal dependencies in dynamical system models and integrates Stein variational gradient descent (SVGD) into a Viterbi-style dynamic programming algorithm, enabling computationally efficient MAP sequence estimation. \textsf{Stein-MAP-Seq} achieves a computational complexity of $\mathcal{O}(M^2)$, where $M$ is the number of particles, in contrast to the $\mathcal{O}(N^2)$ complexity of conventional MAP sequence estimators, with $N \gg M$. Furthermore, the method inherits SVGD's parallelism, enabling efficient computation for real-time deployment on GPU-equipped autonomous systems. We validate the proposed method in various multimodal scenarios, including those arising from nonlinear dynamics with ambiguous observations, unknown data associations, and temporary unobservability, demonstrating substantial improvements in estimation accuracy and robustness to multimodality over existing approaches.