Unifying Sum-Product Networks and Submodular Fields (Supplementary Material)

An SSPN defines an SPN containing a sum node for each possible region of each nonterminal, a product node for each segmentation of each production of each possible region of each nonterminal, and a leaf function on the pixels of the image for each possible region of the image for each terminal symbol. The children of the sum node s for nonterminal Xs with region Rs are all product nodes r with a production vr : Xs → Y1 . . . Yk and region Rvr = Rs. Each product node corresponds to a labeling yr of Rvr and the edge to its parent sum node has weight exp(−wv−E(yr ,Rvr )). The children of product node r are the sum or leaf nodes with matching regions that correspond to the constituent nonterminals or terminals of vr, respectively. Note that this underlying SPN is decomposable, but not smooth. However, (Friesen & Domingos, 2016) showed that smoothness was not a necessary condition for tractable inference and that no corrective factor is necessary when operating in the min-sum semiring, which is what is used for finding the (approximate) optimal parse of an SSPN.