Question $2($Data Selection for Contrastive Learning$)$

Given the following data and latent classes, select a subset of size $4$ using SAS.

Compare similarity between features using negative L$1-$Norm for simplicity. Thus, the optimal subset $S_{**}$ is defined here as:

$S_{**}=argmi{n}_S{\displaystyle \underset{\text{k}\text{i}\text{n}\text{K}}{\overset{?}{}}}{\displaystyle \underset{iin{S}_k}{\overset{?}{}}}{\displaystyle \underset{\text{j}\text{i}\text{n}\frac{V_k}{\frac{?}{?}}{S}_k}{\overset{?}{}}}||x_i-x_j|{|}_1$

where $k$ refers to the latent class of the examples

$V_k$ refers to the set of examples in latent class $k$ in the full data

$S_k$ refers to the set of examples in latent class $k$ in the selected subset

$S={?}_{\text{k}\text{i}\text{n}\text{K}}{S}_k$

Find $S_{**}$ i$.$e$.$ the subset of size $4$ that minimizes the above objective function.

$K=\{0,1\}$

$V_0=\{2,3,4,5,7,9\}$

$V_1=\{1,6,8,10,11,12\}$

The examples are as follows:

$x_1=[0\mathrm{.}0,0\mathrm{.}0,1\mathrm{.}5,3\mathrm{.}0,-0\mathrm{.}05],$ its latent class $=1$

$x_2=[4\mathrm{.}0,6\mathrm{.}0,0\mathrm{.}0,0\mathrm{.}0,0\mathrm{.}05],$ its latent class $=0$

$x_3=[6\mathrm{.}0,4\mathrm{.}0,0\mathrm{.}0,0\mathrm{.}0,-0\mathrm{.}05],$ its latent class $=0$

$x_4=[6\mathrm{.}0,6\mathrm{.}0,0\mathrm{.}0,0\mathrm{.}0,-0\mathrm{.}05],$ its latent class $=0$

$x_5=[8\mathrm{.}0,6\mathrm{.}0,0\mathrm{.}0,0\mathrm{.}0,-0\mathrm{.}05],$ its latent class $=0$

$x_6=[0\mathrm{.}0,0\mathrm{.}0,1\mathrm{.}5,4\mathrm{.}5,-0\mathrm{.}05],$ its latent class $=1$

$x_7=[6\mathrm{.}0,6\mathrm{.}0,0\mathrm{.}0,0\mathrm{.}0,0\mathrm{.}05],$ its latent class $=0$

$x_8=[0\mathrm{.}0,0\mathrm{.}0,1\mathrm{.}5,6\mathrm{.}0,0\mathrm{.}05],$ its latent class $=1$

$x_9=[6\mathrm{.}0,8\mathrm{.}0,0\mathrm{.}0,0\mathrm{.}0,0\mathrm{.}05],$ its latent class $=0$

$x_{10}=[0\mathrm{.}0,0\mathrm{.}0,1\mathrm{.}0,4\mathrm{.}5,0\mathrm{.}05],$ its latent class $=1$

$x_{11}=[0\mathrm{.}0,0\mathrm{.}0,2\mathrm{.}0,4\mathrm{.}5,-0\mathrm{.}05],$ its latent class $=1$

$x_{12}=[0\mathrm{.}0,0\mathrm{.}0,1\mathrm{.}5,4\mathrm{.}5,0\mathrm{.}05],$ its latent class $=1$

Select indices of all examples that will be selected.