In the paper, we addressed that "weighted" LCNN is a convex function but after we found that it is not the case in general (uniform weight cases are convex). Theoretically, the (nonuniformly) weighted LCNN becomes a convex function only in the case where the right and left singular vectors corresponding to $\sigma_1$ are fixed. Fortunately, the algorithm proposed in the paper shows a stable convergence in practice even using the weighted LCNN. This might be because the first right-and-left singular vectors of each local color matrix, namely, a rough direction of the color line, is almost fixed across iterations.
We apologize for any confusion this may have caused.