Author: Mazuran
Year: 2014
- Inference on factor graph is equivalent to NLS optimization
- Formulation of the good optimization pb of marginalization and sparsification:
- pb convex
- conserve block structure
- general approach
- local approach (only in markov blanket)
- demonstration of KLD being an instance of MAXDET problem, which makes it a convex problem
- optimzation based on limited memory Projected Quasi Newton (PQN):
- adaptation of Proximal Quasi Newton to box constraints pb (case of KLD minimization)
- define a set of variables that are too close to boundaries and fix them
- define the Hessian block of free variables
- take a projected Newton step
- Here the projection takes the closest positive definite matrix, in the case of symmetric matrices, there exists a closed form solution $$ P(X) = \underset{Y \succ 0}{\argmin} ||X-Y||_F^2 = V diag(max(0, \lambda_i)) V^T $$
Application to 2D graph SLAM:
- assume that they know which node to remove
- steps of marginalization and sparsification:
- extract Markov Blanket of a node
- Optimize node to get a local linearization point
- Marginalize the point using Schur Complement
- Select a set of virtual measurments
- Solve KLD
- If
$\Omega$ is not invertible, it means that Markov Blanket lacks a rigid transform edge to the World Frame => needs to project it on a subspace where it is invertible - Select the Topology using Chow Liu Tree for the tree based approximation
- Chow Liu Tree + greedy heuristic for the subgraph based approximation
Experiments:
- Manhattan and Intel are full 2D datasets
- Try 4 levels of node reduction
- what is "fill in" ? percentage of non zero elements?