  Modern depth sensors can generate a huge number of 3D points in few seconds
to be latter processed by Localization and Mapping algorithms. Ideally, these
algorithms should handle efficiently large sizes of Point Clouds under the
assumption that using more points implies more information available. The Eigen
Factors (EF) is a new algorithm that solves SLAM by using planes as the main
geometric primitive. To do so, EF exhaustively calculates the error of all
points at complexity $O(1)$, thanks to the {\em Summation matrix} $S$ of
homogeneous points.
  The solution of EF is highly efficient: i) the state variables are only the
sensor poses -- trajectory, while the plane parameters are estimated previously
in closed from and ii) EF alternating optimization uses a Newton-Raphson method
by a direct analytical calculation of the gradient and the Hessian, which turns
out to be a block diagonal matrix. Since we require to differentiate over
eigenvalues and matrix elements, we have developed an intuitive methodology to
calculate partial derivatives in the manifold of rigid body transformations
$SE(3)$, which could be applied to unrelated problems that require analytical
derivatives of certain complexity.
  We evaluate EF and other state-of-the-art plane SLAM back-end algorithms in a
synthetic environment. The evaluation is extended to ICL dataset (RGBD) and
LiDAR KITTI dataset. Code is publicly available at
https://github.com/prime-slam/EF-plane-SLAM.
