This paper describes serial and parallel compositional models of multiple objects with part sharing. Objects are built by part-subpart compositions and expressed in terms of a hierarchical dictionary of object parts. These parts are represented on lattices of decreasing sizes which yield an executive summary description. We describe inference and learning algorithms for these models. We analyze the com- plexity of this model in terms of computation time (for serial computers) and num- bers of nodes (e.g., ”neurons”) for parallel computers. In particular, we compute the complexity gains by part sharing and its dependence on how the dictionary scales with the level of the hierarchy. We explore three regimes of scaling be- havior where the dictionary size (i) increases exponentially with the level, (ii) is determined by an unsupervised compositional learning algorithm applied to real data, (iii) decreases exponentially with scale. This analysis shows that in some regimes the use of shared parts enables algorithms which can perform inference in time linear in the number of levels for an exponential number of objects. In other regimes part sharing has little advantage for serial computers but can give linear processing on parallel computers.
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