Expectation of the inner product of a subset of two random orthonormal vectors

Question

Setting: Consider sampling two orthonormal vectors $\mathbf{u},\mathbf{v}$ in $\mathbb{R}^p$ (where $p\ge2$) from a "uniform" distribution over the $p$-dimensional sphere (alternatively, sample $\mathbf{u},\mathbf{v}\sim\mathcal{S}^{p-1}$ and then orthogonalize and normalize $\mathbf{v}$).

Let $\lceil p/2\rceil\le m\le p$.
Question: Can we upper bound the following expectation? $$ \mathbb{E}_{\substack{\mathbf{u},\mathbf{v}\sim\mathcal{S}^{p-1}:\\\mathbf{u}\perp\mathbf{v}}} \left(\mathbf{u}^\top\left[\begin{array}{cc} \mathbf{I}_{m}\\ & \mathbf{0}_{p-m} \end{array}\right]\mathbf{v}\right)^2 = \mathbb{E}_{\substack{\mathbf{u},\mathbf{v}\sim\mathcal{S}^{p-1}:\\\mathbf{u}\perp\mathbf{v}}} \left(\sum_{i=1}^{m}u_iv_i\right)^2 $$

Example: when $m$ is very close to $p$, the expectation should be very small, since the vectors are nearly orthogonal.

Direction: We thought about approximating this with just two independent Gaussian vectors and look for the concentration bounds when $p\to\infty$, but this seems perhaps too loose?

Observation: Since the vectors are orthogonal, we have $ \left(\mathbf{u}^\top\left[\begin{array}{cc} \mathbf{I}_{m}\\ & \mathbf{0}_{p-m} \end{array}\right]\mathbf{v}\right)^2 = \left(\mathbf{u}^\top\mathbf{v}-\mathbf{u}^\top\left[\begin{array}{cc} \mathbf{I}_{m}\\ & \mathbf{0}_{p-m} \end{array}\right]\mathbf{v}\right)^2 = \left(\mathbf{u}^\top\left[\begin{array}{cc} \mathbf{0}_{m}\\ & \mathbf{I}_{p-m} \end{array}\right]\mathbf{v}\right)^2$, and thus we can equivalently focus on $0\le m\le \lfloor p/2\rfloor$.

Any help and ideas would be greatly appreciated!

Answer 1

Denote $\alpha=\mathbb{E} u_1^2v_1^2$, $\beta=\mathbb{E} u_1v_1u_2v_2$. Then by the symmetry and linearity of expectation we have $$f(m):=\mathbb{E} (u_1v_1+\ldots+u_mv_m)^2=m\alpha+m(m-1)\beta.$$ We have $f(p)=0$, thus $\beta=-\alpha/(p-1)$, and $f(m)=\alpha m(p-m)/(p-1)$.

It remains to bound $\alpha$. Choose a vector $w=(w_1,\ldots,w_p)\in \mathcal{S}^{p-1}$ independent of $u,v$, then $\alpha=\mathbb{E} \langle u,w\rangle^2\cdot \langle v,w\rangle^2$, as the conditional expectation clearly does not depend on $w$ (here $\langle \cdot,\cdot\rangle$ stands for the inner product). On the other hand, the conditional expectation does not depend on the pair $u,v$ of orthogonal vectors $u,v$, thus we may take $u=(1,0,\ldots,0)$, $v=(0,1,\ldots,0)$, and $\alpha=\mathbb{E} w_1^2w_2^2$.

Next, we have $$1=\mathbb{E} \left(\sum_{i=1}^p w_i^2\right)^2=\alpha\cdot p(p-1)+p\cdot \mathbb{E} w_1^4,$$ thus $$\alpha=\frac{1-p\cdot \mathbb{E} w_1^4}{p(p-1)}.$$ To find $\mathbb{E} w_1^4$, we may think that $w=(w_1,\ldots,w_p)=(\xi_1,\ldots,\xi_p)/\sqrt{\sum \xi_i^2}$ where $\xi_i$ are i.i.d. standard normal. Then $$ \mathbb{E} w_1^4=\mathbb{E} \frac{\xi_1^4}{(\sum \xi_i^2)^2}=:\Theta $$ By some form of law of large numbers like Chernoff bound, the probability that $\sum \xi_i^2<p/2$ is exponentially small in $p$, that gives exponentially small contribution to the expectation $\Theta$. If $\sum \xi_i^2>p/2$, then $$\frac{\xi_1^4}{(\sum \xi_i^2)^2}<\frac{4}{p^2}\xi_1^4,$$ and since $\mathbb{E} \xi_1^4$ is a finite constant, we conclude that $\Theta=O(1/p^2)$. Thus $\alpha=1/p^2+O(1/p^3)$.

Answer 2

This is to complement the nice answer by Fedor Petrov by providing the exact expression for his $\alpha=Ew_1^2w_2^2$ and thus for the desired expectation $f(m)=\alpha m(p-m)/(p-1)$.

Just note that the random vector $(w_1^2,\dots,w_p^2)$ coincides in distribution with the random vector $\dfrac{(G_1^2,\dots,G_p^2)}{G_1^2+\dots+G_p^2}$, where the $G_i$'s are independent standard normal random variables, and hence the joint distribution of $w_1^2$ and $w_2^2$ is the Dirichlet distribution with parameters $1/2,1/2,p/2-1$. Therefore, after some simple calculations we get $$\alpha=Ew_1^2w_2^2=\frac1{p(p+2)}$$ (which is in agreement with Fedor Petrov's conclusion that $\alpha=1/p^2+O(1/p^3)$).