Exploring Compositional High Order Pattern Potentials for Structured Output Learning Supplementary Material

This section provides the detailed proof of the equivalence between pattern potentials and RBMs. The high level idea of the proof is to treat each hidden variable in an RBM as encoding a pattern. We first introduce the definition of pattern potentials by Rother et al. in [2], a few necessary change of variable tricks, and two different ways to compose more general high order potentials, “sum” and “min”. Then we relates the composite pattern potentials to RBMs. We show in Section 1.2 that minimizing out hidden variables in RBMs are equivalent to pattern potentials. When there are no constraints on hidden variables, we recover the “sum” composite pattern potentials; when there is a 1-of-J constraint on hidden variables, we recover the “min” composite pattern potentials. In Section 1.3, we show that summing out hidden variables in RBMs approximates pattern potentials, and similarly with and without constraints on hidden variables would lead us to “min” and “sum” cases respectively. The RBM formulation offers considerable generality via choices about how to constrain hidden unit activations. This allows a smooth interpolation between the “sum” and “min” composition strategies. Also, this formulation allows the application of learning procedures that are appropriate for cases other than just the “min” composition strategy. In Section 2, we provide a way to unify minimizing out hidden variables and summing out hidden variables by introducing a temperature parameter in the model. Notation. In this section, we use g for pattern potentials and ĝ for the high order potentials induced by RBMs. Superscripts ‘s’ and ‘m’ on g corresponds to two composition schemes, sum and min. Superscripts on ĝ correspond to two types of constraints on RBM hidden variables, and subscripts on ĝ correspond to minimizing out or summing out hidden variables.