Random rotations
May 17, 2021 — December 1, 2021
Placeholder for random measures on the special orthogonal group, which encodes random
1 Uniform random rotations
Key word: Haar measure.
Chatterjee and Meckes (2008) gives us a useful lemma describing the moments of a random rotation:
If
- For all
, - For all
,
Let
2 Tiny random rotations
Notation:
2.1 Random Givens rotation
Givens rotations are useful. We can parameterize a Givens rotation
For
Informally (because I do not have space or will to track lots of boring remainder terms) we note that for some fixed
If we want this to work in a higher dimension, we pad it like this:
Now, let us consider how to randomise these. If we fix two axes and a random
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
2.3 Random rotation in an isotropic direction
Suppose
Then
Once again referring to Chatterjee and Meckes (2008) we see that the matrix
\[\begin{aligned}
\mathrm{Q}^{\varepsilon}-\mathrm{I}
&\approx
\mathrm{Q}\left(
\mathrm{I}_{d}
+\left[
TBC.
3 Sparse
We could desire a random rotation be sparse in a few senses, e.g. “ a random rotation in
A useful context for me is to take something that is expected to be a rotation, but whose realisations are sparse.