A hard thresholding operator $H_k:\mathbb{R}^n\rightarrow \mathbb{R}^n$ is defined as a vector-valued function that maintains the top-k entries of a given vector in an absolute value sense and zero out the rest. As an example $H_2(x)=[-5,0,-3,0]^{\top}$ where $x=[-5,2,-3,1]^{\top}$ and $k=2$. According to Is hard thresholding operator Lipschitz? , we know that it is not Lipschitz.
I have a situation where arguments of hard thresholding operator are very close to each other, that is, $y_1=x$ and $y_2=x+\xi$ where $\xi$ is a random vector in $\mathbb{R}^n$ whose entries are normal random variable with zero mean and variance $\sigma$ which can be reduced as small as possible but not zero.
Question: Is there anyway to simplify $||H_k(y_1)-H_k(y_2)||_2$ and have an expression in terms of $y_1, y_2$?
Would it be possible to impose locally Lipschitzness condition given $y_1$ and $y_2$?
Can we exploit the bound found in A Tight Bound of Hard Thresholding to find a bound on $||H_k(y_1)-H_k(y_2)||_2$ in terms of $y_1, y_2$?
Another view: In essence, what I am asking is that when $\mathbb{E}[x+\xi]=x$, would it be possible to say something about the following:
$$ \mathbb{E}[||H_k(x)-H_k(x+\xi)||_2] $$
or
$$ ||H_k(x)-\mathbb{E}[H_k(x+\xi)]||_2 $$