Algorithm 1 LDL with Multi-scale Neighborhood Granule

1:Input: Feature matrix X, label matrix Y, feature subset F, neighborhood thresholds $\Theta =\{{\theta }_1,\dots ,{\theta }_K\}$   2:Output: Multi-scale label distribution matrix $D\in {\mathbb{R}}^{n\times q}$   3:for each scale ${\theta }_k\in \Theta$ do   4:      for each instance $x_i\in X$ do   5:            Compute neighborhood granule ${\mathcal{N}}_F^{{\theta }_k}\left(x_i\right)=\{x_h\mid {\Delta }_F(x_i,x_h)\le {\theta }_k\}$   6:            for each label $l_j\in Y$ do   7:                  Compute label distinguishability $D_{i,j}^{{\theta }_k}$ by (8)   8:                  Compute Shannon entropy $H_{i,j}^{{\theta }_k}$ by (9)   9:            end for 10:      end for 11:      Compute average neighborhood entropy: $H^{{\theta }_k}={\displaystyle \frac{1}{nq}}{\sum }_{i=1}^n{\sum }_{j=1}^q{H}_{i,j}^{{\theta }_k}$ 12:end for 13:Compute scale weight: $w^{{\theta }_k}={\displaystyle \frac{\exp (-H^{{\theta }_k})}{{\sum }_{r=1}^K\exp (-H^{{\theta }_r})}}$ 14:for each instance $x_i$ and label $l_j$ do 15:      Fuse multi-scale label distribution: $D_{i,j}={\sum }_{k=1}^K{w}^{{\theta }_k}{D}_{i,j}^{{\theta }_k}$ 16:end for 17:row-wise normalization: $D_{i,j}={\displaystyle \frac{D_{i,j}}{{\sum }_{j=1}^q{D}_{i,j}}},j=1,2,\dots ,q$ 18:Return: Multi-scale label distribution matrix D