# Question about minimax risk sparse sparse condition

I am studying statistical leaning theory.

Especially the paper "Minimax rates of Estimation for High Dimensional Linear regression over $l_q$ balls" by Garvesh Raskutti .et.al.

In the right end of the page 6990 (proof of lemma 4)

there is the situation below

$H=\left\{−1,0,1\right\}^d$ => $d$ dimensional vector whose elements is one of $\left\{-1,0,1\right\}$

$H^∗=\left\{v∈h:||v||_o =s\right\}$

$||v||_o$ is the number of nonzero elements of $v$

that is $H^*$ is the set of all $s$ sparse vector of $d$ dimension

and for $v\in H^*$ , $B(v,s/2)=\left\{v ′ ∈H^∗ :p(v,v')≤s/2\right\}$,

$p(v,v′)$ is hamming distance between $v,v′$, that is $B(v,s/2)$

$d≥3s/2$

I need to prove $|B(v,s/2)|≤$ ${d}\choose{s/2}$ $2^(s/2)$.

At this paper the proof is not presented.

$H=\left\{−1,0,1\right\}^d$

=> the set of all $d$ dimensional vector whose elements is one of {−1,0,1}

$H^∗ =\left\{v∈H:||v||_o =s\right\}$

• Please register &/or merge your accounts (you can find information on how to do this in the My Account section of our help center), then you will be able to edit & comment on your own question. Jul 10, 2018 at 6:03

$H=\left\{−1,0,1\right\}^d$
=> the set of all $d$ dimensional vector whose elements is one of {−1,0,1}