Supplementary Material: Learning Affinity via Spatial Propagation Networks

Theorem 1 (i.e., Theorem 3 in the paper) shows that the stability of a linear propagation model can 8 be maintained by regularizing all the weights of each pixel in the hidden layer such the summation 9 of their absolute values is less than one. For the one-way connection, Chen et al. [1] maintain each 10 scalar output p to be within (0, 1). Liu et al. [4] extend the range to (−1, 1), where the negative 11 weights show preferable effects for learning image enhancers. This indicates that the affinity matrix 12 is not necessarily restricted to be positive/semi-positive definite. (e.g., this setting is also used for 13 a pre-defined affinity matrix in [3].) For the three-way connection, we simply regularize the three 14 weights (the output of a deep CNN) according to Theorem 1 without any positive/semi-positive 15 definite restriction. 16