Fig. 7: CSCGs enable hierarchical abstraction and planning.
The cloned graph of the CSCG lifts the observations into a hidden space, allowing for discovery of modularity that is not apparent in the visible observations. a The modular graph from8, modified to have aliased observations. Observations at each node are indicated by the numbers, and many different nodes produce the same observation. b MDS or community detection on the SR matrix of random walks in a does not reveal the modularity of the graph. c Community detection on the CSCG transition matrix successfully recovers the modularity of graph in a, recovering three communities. d A maze that has an embedded three-level hierarchy. Sensory observations are aliased both within rooms and across rooms. The black pixels denote “bridges” between the rooms. CSCG is trained on random walks from this maze. Community detection on the learned CSCG transition matrix revealed a first level of organization into rooms (e), and another level of community detection revealed hyper-rooms (f), resulting in a three-level hierarchical graph reflecting the nested structure of the maze. Planning a path (black arrows) between two rooms (denoted as (S)tart (filled black dot) and (F)inish (open black dot) in d) was achieved by finding the shortest path between hyper-rooms to navigate, next finding the shortest path between rooms, and lastly finding the shortest path within the rooms in this reduced search space. g Visualization of planning message propagation in the one-level graph. Messages propagate in the whole maze, indicating a wide search area. h Visualization of hierarchical planning. Routes are first identified on the highest level, which then becomes sub-goals at the lower level. The red colored nodes indicate the sequence of sub-goals, and their intensities reflect the ordering of the sub-goals. Compared to g, hierarchical planning requires fewer messages to be propagated so it is faster.
