# Compression Bounds - Determine and Visualize for hypothesis vs VC dimension

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 $$H_{decst}$$.

Generally, every class of finite VC dimension admits to a (exponential size in the dimension) compression scheme. This was an open question resolved a few years ago by Shay and Amir in this paper. However, this is an overkill for your question, since $$H_k$$ obviously has a $$k$$-comprresion scheme (only keep the samples labeled $$1$$, of which there are at most $$k$$, and the reconstruction is zero everywhere else).
$$H_{desct}$$ can be compressed to two samples, given $$\left((a_i,b_i),y_i\right)_{i=1}^n$$, we know that either there exists $$j$$ such $$\forall i\in[n]: y_i=\mathbb{1}_{a_i\le a_j}$$ or there exists $$j$$ such that $$\forall i\in[n] : y_i=\mathbb{1}_{b_i\le b_j}$$. The compression keeps $$(a_j,b_j)$$ and one more sample $$(a_{j'},b_{j'})$$ to indicate the correct coordinate of the threshold, e.g. if the former condition holds and $$\forall i\in[n]: y_i=\mathbb{1}_{a_i\le a_j}$$ then pick $$j'$$ such that $$y_{j'}\neq y_j$$ (so $$y_{j'}=0)$$ and $$b_{j'} (this allows us to reconstruct the function $$\mathbb{1}_{x_1\le a_j}$$ from the two samples). If no such $$j'$$ exists then $$\mathbb{1}_{x_2\le b_j}$$ is also a consistent hypothesis, and our compression can keep only one sample.
• Do you mean $\mathcal{X}=2^{\mathbb{N}}$ and $\mathcal{H}=\{h_{x_0} | x_0\in\mathbb{N}\}$ where $h_{x_0}(A)=\mathbb{1}_{x_0\in A}$? In that case $VC_{\mathcal{H}}=\infty$ (Given $n\in\mathbb{N}$ construct an $n$-size shattered set as follows, select different $x_1,...,x_{2^n}\in\mathbb{N}$ and index the subsets of $[n]$ by $1,...,2^n$. Now simply generate sets $A_1,...,A_n$ such that if $I\subseteq[n]$ is the $i'th$ subset then $x_i\in A_j$ iff $j\in I$. You can show that this set is indeed shattered). Infinite VC dimension implies no constant size compression scheme. Dec 14 '20 at 7:22