A Proof for the Approximate Sparsity of SLAM Information Matrices

For the Simultaneous Localization and Mapping problem several efficient algorithms have been proposed that make use of a sparse information matrix representation (e.g. SEIF, TJTF, treemap). Since the exact SLAM information matrix is dense, these algorithm have to approximate it (sparsification). It has been empirically observed that this approximation is adequate because entries in the matrix corresponding to distant landmarks are extremely small. This paper provides a theoretical proof for this observation, specifically showing that the off-diagonal entries corresponding to two landmarks decay exponentially with the distance traveled between observation of first and second landmark.