Constrained Submodular Minimization Towards Missing Labels and Class Imbalance in Multi-label Learning

Copyright c © 2016, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. edges among instances EX for each class (building connections between the entries within the same row, for every row of Z); last, we copy the edges among classes EC for each instance (building connections between the entries within the same column, for every column of Z). di 6= ∑mn j W(i, j): if (i, j) ∈ EX , then di = dX (̂i), with î being the instance index (i.e., the column index of Z) corresponding to the node i in G; similarly, if (i, j) ∈ EC , then di = dC (̂i). That means we normalize W(i, j) by the sum of instance-level neighbors (in the same column) or class-level neighbors (in the same row), rather than the sum of all neighbors. As a result, this problem is a partially normalized graph-cut problem. Interestingly, the formulation in (3) is exactly the same as that of the standard GSSL problem (Zhu 2006). The only difference is that L is not a normalized Laplacian matrix in (3). Please see Lemma 1.