# General Purpose Incremental Covariance Update and Efficient Belief Space Planning via Factor-Graph Propagation Action Tree

## TL;DR

This paper introduces a novel incremental covariance update method that accelerates belief space planning by reusing computations across similar candidate actions, significantly improving efficiency in high-dimensional SLAM and planning tasks.

## Contribution

The paper presents a general-purpose incremental covariance update technique and an efficient belief space planning approach that leverages covariance re-use for similar candidate actions.

## Key findings

- Faster covariance recovery compared to state-of-the-art methods.
- Efficient belief space planning with reduced computation for similar actions.
- Extension of previous BSP method to exploit incremental covariance updates.

## Abstract

Fast covariance calculation is required both for SLAM (e.g.~in order to solve data association) and for evaluating the information-theoretic term for different candidate actions in belief space planning (BSP). In this paper we make two primary contributions. First, we develop a novel general-purpose incremental covariance update technique, which efficiently recovers specific covariance entries after any change in the inference problem, such as introduction of new observations/variables or re-linearization of the state vector. Our approach is shown to recover them faster than other state-of-the-art methods. Second, we present a computationally efficient approach for BSP in high-dimensional state spaces, leveraging our incremental covariance update method. State of the art BSP approaches perform belief propagation for each candidate action and then evaluate an objective function that typically includes an information-theoretic term, such as entropy or information gain. Yet, candidate actions often have similar parts (e.g. common trajectory parts), which are however evaluated separately for each candidate. Moreover, calculating the information-theoretic term involves a costly determinant computation of the entire information (covariance) matrix which is O(n^3) with n being dimension of the state or costly Schur complement operations if only marginal posterior covariance of certain variables is of interest. Our approach, rAMDL-Tree, extends our previous BSP method rAMDL, by exploiting incremental covariance calculation and performing calculation re-use between common parts of non-myopic candidate actions, such that these parts are evaluated only once, in contrast to existing approaches.