I want to visualize or calculate the compression bounds for hypothesis classes. I learnt how to figure out the VC dimension. Let's say I define two hypothesis class. For example: $$ H_k = \{h ∈ (0,1)_X \mid |h^{−1}(1)| ≤ k\} $$
and $$ H_{decst} = \{h^i_a \mid a ∈ R, i ∈ (1,2)\}, \text{ where } h^i_a((x1,x2)) = 1[x_i ≤ a]. $$
For VC dimension, I showed The VC dimension of $H_{decst}$ is 2. We can shatter the two points $x_1 = (2,0)$ and $x_2 = (0, 2)$ with the four functions $h_{0.5}, h_{1.5}, h_{1.5}, h_{2.5}$.
Also, It is not hard to see that $H_k$ shatters any set $C ⊆ N$ of size $k$, but no larger set, since the all-1 labeling can not be realized on more than $k$ points with $H_k$. Thus, the VC-dimension of $H_k$ is $k$.
Similarly, I want to figure out the compression bounds for $H_k$ and $H_{decst}$.