Distance Metric, Heurisitic Function

Manhattan Distance (L1 Loss)

The Manhattan Distance is given by $|x_f-x_i|+|y_f-y_i|$ This notation is used for Computer Vision: $d_1(I_1,I_2)=\sum _p|I_1^p-I_2^p|$

Resources

• https://pytorch.org/docs/stable/generated/torch.nn.L1Loss.html#torch.nn.L1Loss

$\ell (x,y)=L=\{l_1,\dots ,l_N{\}}^{\top },l_n=|x_n-y_n|$

• where $N$ is the batch size (not the number of batches)

$$\ell (x,y)=\{\begin{array}{ll}\mathrm{mean}(L),& \text{if reduction = ’mean’};\\ \mathrm{sum}(L),& \text{if reduction = ’sum’}.\end{array}$$

• So $\ell (x,y)$ is the loss of a single batch
• The actual reduction has two options, either mean or sum (default is mean)