Supplementary Material: Distribution Aligning Refinery of Pseudo-label for Imbalanced Semi-supervised Learning A Proof of Theorem 1, r 2 R n 0, c 2 R m 0

In this section, we present the formal proof of Theorem 1. To this end, we interpret DARP as a coordinate ascent algorithm of the Lagrangian dual of its original objective (1), and discuss the necessary and sufficient condition of correct convergence of DARP, i.e., convergence to the optimal solution of (1). Now, we will show that DARP is indeed a coordinate ascent algorithm for the dual of the above optimization. To this end, we formulate the Lagrangian dual of (3). In addition, the optimal objective value of (3) is equivalent to that of (4), i.e., the strong duality holds.