## Complexity of Representation and Inference in Compositional Models with Part Sharing

*Alan Yuille, Roozbeh Mottaghi*; 17(11):1−28, 2016.

### Abstract

This paper performs a complexity analysis of a class of serial and parallel compositional models of multiple objects and shows that they enable efficient representation and rapid inference. Compositional models are generative and represent objects in a hierarchically distributed manner in terms of parts and subparts, which are constructed recursively by part-subpart compositions. Parts are represented more coarsely at higher level of the hierarchy, so that the upper levels give coarse summary descriptions (e.g., there is a horse in the image) while the lower levels represents the details (e.g., the positions of the legs of the horse). This hierarchically distributed representation obeys the *executive summary* principle, meaning that a high level executive only requires a coarse summary description and can, if necessary, get more details by consulting lower level executives. The parts and subparts are organized in terms of hierarchical dictionaries which enables part sharing between different objects allowing efficient representation of many objects. The first main contribution of this paper is to show that compositional models can be mapped onto a parallel visual architecture similar to that used by bio- inspired visual models such as deep convolutional networks but more explicit in terms of representation, hence enabling part detection as well as object detection, and suitable for complexity analysis. Inference algorithms can be run on this architecture to exploit the gains caused by part sharing and executive summary. Effectively, this compositional architecture enables us to perform exact inference simultaneously over a large class of generative models of objects. The second contribution is an analysis of the complexity of compositional models in terms of computation time (for serial computers) and numbers of nodes (e.g., "neurons") for parallel computers. In particular, we compute the complexity gains by part sharing and executive summary and their dependence on how the dictionary scales with the level of the hierarchy. We explore three regimes of scaling behavior where the dictionary size (i) increases exponentially with the level of the hierarchy, (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 enable linear processing on parallel computers.

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