Random rotations

2.1 Random Givens rotation

Givens rotations are useful. We can parameterize a Givens rotation ${\mathrm{A}}^{(t)}$ by either an angle $t=\theta$ or a magnitude $t=ϵ=\sin \theta$. The latter is slightly easier. Formally, a Givens rotation can be defined in any dimension $d\ge 2$ by padding it with zeros and adding it to an identity matrix, but all the action happens over 4 entries, so we can learn how it works by examining a minimal $2\times 2$ rotation as a perturbation of the identity.

For $-1\le t\le 1$ fixed, let us define a rotation $$\begin{array}{rl}{\mathrm{A}}^{(t)}& =\left[\begin{array}{cc}\sqrt{1-t^2}& t\\ -t& \sqrt{1-t^2}\end{array}\right]\\ & ={\mathrm{I}}_2+\left[\begin{array}{cc}-\frac{t^2}{2}+\delta & t\\ -t& -\frac{t^2}{2}+\delta \end{array}\right]\end{array}$$ where $\delta$ is a deterministic constant and $\delta =O\left(t^4\right).$

Informally (because I do not have space or will to track lots of boring remainder terms) we note that for some fixed $x=[x_1{x}_2{]}^{\mathrm{\top }}$ we have $$\begin{array}{rl}{\mathrm{A}}^{(t)}x{x}^{\mathrm{\top }}{\mathrm{A}}^{(t)}{}^{\mathrm{\top }}& \approx \left(\begin{array}{cc}{x}_1^2+2t{x}_1{x}_2+t^2(-x_1^2+x_2^2)& x_1{x}_2-2t^2{x}_1{x}_2+t(-x_1^2+x_2^2)\\ x_1{x}_2-2t^2{x}_1{x}_2+t(-x_1^2+x_2^2)& -2t{x}_1{x}_2+x_2^2+t^2(x_1^2-x_2^2)\end{array}\right)\\ & =x{x}^{\mathrm{\top }}+\left(\begin{array}{cc}2x_1{x}_2& -x_1^2+x_2^2\\ -x_1^2+x_2^2& -2x_1{x}_2\end{array}\right)t+\left(\begin{array}{cc}-x_1^2+x_2^2& -2x_1{x}_2\\ -2x_1{x}_2& x_1^2-x_2^2\end{array}\right)t^2\end{array}$$

If we want this to work in a higher dimension, we pad it like this: $$\begin{array}{rl}{\mathrm{A}}^{(t)}& =\left[\begin{array}{cc}\sqrt{1-t^2}& t\\ -t& \sqrt{1-t^2}\end{array}\right]\oplus {\mathrm{I}}_{d-2}\\ & ={\mathrm{I}}_d+\left[\begin{array}{cc}-\frac{t^2}{2}+\delta & t\\ -t& -\frac{t^2}{2}+\delta \end{array}\right]\oplus 0_{d-2}\end{array}$$

Now, let us consider how to randomise these. If we fix two axes and a random $t$ we can construct a random Givens rotation. Suppose $\mathbb{E}[t]=0$, $\mathrm{var}(t)={\epsilon }^2$ small, and proceeding to do a sloppy Taylor expansion (which will, I assert, not matter in practice when take limits as $t\to 0$ but one should check this to be rigorous). $$\begin{array}{rl}\mathbb{E}[{\mathrm{A}}^{(t)}x]& \approx \mathbb{E}\left[\left(\begin{array}{c}{x}_1+t{x}_2-\frac{t^2{x}_1}{2}\\ x_2-t{x}_1-\frac{t^2{x}_2}{2}\end{array}\right)\right]\\ & =x-\mathbb{E}[t^2]x.\end{array}$$

$$\begin{array}{rl}\mathbb{E}[{\mathrm{A}}^{(t)}x{x}^{\mathrm{\top }}{\mathrm{A}}^{(t)}{}^{\mathrm{\top }}]& \approx \mathbb{E}[x{x}^{\mathrm{\top }}+\left(\begin{array}{cc}2x_1{x}_2& -x_1^2+x_2^2\\ -x_1^2+x_2^2& -2x_1{x}_2\end{array}\right)t+\left(\begin{array}{cc}-x_1^2+x_2^2& -2x_1{x}_2\\ -2x_1{x}_2& x_1^2-x_2^2\end{array}\right)t^2]\\ & =x{x}^{\mathrm{\top }}+\left(\begin{array}{cc}-x_1^2+x_2^2& -2x_1{x}_2\\ -2x_1{x}_2& x_1^2-x_2^2\end{array}\right)\mathbb{E}[t^2]\end{array}$$

2.2 Doubly Random Givens rotation

If each Givens rotation is over a pair of distinct axes drawn uniformly at random, this will also give us a random rotation. Bookkeeping for these could be irritating, but let us see. wlog we can generate these by choosing the axes $i\sim \mathcal{U}(\{1,2,\dots ,n-1\},j\sim \mathcal{U}(\{i+1,i,\dots ,n\}.$

2.3 Random rotation in an isotropic direction

Suppose $\mathrm{Q}$ is a Haar-distributed (i.e. uniform) random rotation. Then we can make a tiny incremental rotation using a construction of Chatterjee and Meckes (2008), by applying a small Given rotation in a randomly rotated coordinate frame. The result is a rotation in a random direction, but not far in that direction.

Then ${\mathrm{Q}}^{\epsilon }:=\mathrm{Q}{\mathrm{A}}_{\epsilon }{\mathrm{Q}}^{\mathrm{\top }}$ is a small random rotation. What can we say about this guy? Let $Q_{1:2}$ be the $d\times 2$ matrix made of the first two columns of $\mathrm{Q}$ and $C_2=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$. Then $${\mathrm{Q}}^{\epsilon }=ϵ[-(\frac{ϵ}{2}+{ϵ}^{-1}\delta )Q_{1:2}{Q}_{1:2}^{\mathrm{\top }}+Q_{1:2}{C}_2{Q}_{1:2}^{\mathrm{\top }}]$$ and ${ϵ}^{-1}\delta =O\left({ϵ}^3\right)$.

Once again referring to Chatterjee and Meckes (2008) we see that the matrix $M={\left[m_{ij}\right]}_{i,j=1}^n=M_{1:2}{C}_2{M}_{1:2}^{\mathrm{\top }}$ satisfies $m_{ij}=u_{i1}{u}_{j2}-u_{i2}{u}_{j1}.$ For all $i,j,\ell ,p$, $$\mathbb{E}\left[m_{ij}{m}_{\ell p}\right]=\frac{2}{n(n-1)}[{\delta }_{i\ell }{\delta }_{jp}-{\delta }_{ip}{\delta }_{j\ell }]$$ By the lemma on expectations of random rotations, $\mathbb{E}\left[Q_{1:2}{Q}_{1:2}^{\mathrm{\top }}\right]=\frac{2}{n}I$.

TBC.