Stein-MAP: A Sequential Variational Inference Framework for Maximum A Posteriori Estimation

State estimation poses substantial challenges in robotics, often involving encounters with multimodality in real-world scenarios. To address these challenges, it is essential to calculate Maximum a posteriori (MAP) sequences from joint probability distributions of latent states and observations over time. However, it generally involves a trade-off between approximation errors and computational complexity. In this article, we propose a new method for MAP sequence estimation called Stein-MAP, which effectively manages multimodality with fewer approximation errors while significantly reducing computational and memory burdens. Our key contribution lies in the introduction of a sequential variational inference framework designed to handle temporal dependencies among transition states within dynamical system models. The framework integrates Stein's identity from probability theory and reproducing kernel Hilbert space (RKHS) theory, enabling computationally efficient MAP sequence estimation. As a MAP sequence estimator, Stein-MAP boasts a computational complexity of O(N), where N is the number of particles, in contrast to the O(N^2) complexity of the Viterbi algorithm. The proposed method is empirically validated through real-world experiments focused on range-only (wireless) localization. The results demonstrate a substantial enhancement in state estimation compared to existing methods. A remarkable feature of Stein-MAP is that it can attain improved state estimation with only 40 to 50 particles, as opposed to the 1000 particles that the particle filter or its variants require.