Average of cosine similarity between $n$-vectors sampled from $d$-dim hypersphere

Question

I'm having a tough time on this problem, suppose we have $v_i$ for $i=1\cdots n$ unit vectors sampled from a $d$-dimensional hypersfere. How can we evaluate this average over these vectors? $$ \mathbb{E}_v\frac{1}{n^4}\left(\sum_{i,j=1}^{n,n}(v_i\cdot v_j)^2\right)^2=? $$ I was able to determine that $$ \mathbb{E}_v\frac{1}{n^2}\sum_{i,j=1}^{n,n}(v_i\cdot v_j)^2 = \frac{(n-1)/d+1}{n} $$ now I also need the fourth moment but I am unable to perform the calculations... does anyone have an idea on how to proceed?

Answer 1

Hint: Note that the sum under the expectation can be written as $$\sum_{i,j,k,l}\frac{(z_i.z_j)^2(z_k.z_l)^2}{\|z_i\|^2\|z_j\|^2\|z_k\|^2\|z_l\|^2},$$ where $z_i$ are standard normal random vectors in $R^d$. You have to now consider the cases where $2,3$ or $4$ of these indices are same, and the corresponding terms will be of the form: $$\frac{(z_i.z_j)^2}{\|z_k\|^2\|z_l\|^2}, or \frac{(z_i.z_j)^2(z_i.z_l)^2}{\|z_i\|^4\|z_j\|^2\|z_l\|^2}, or 1.$$ Now, you can use the rotational invariance of the Gaussians along with their spherical coordinate representation to simplify these expectations and obtain the final result.