On distributions over orthonormal sets: existing families, construction, and simulation

Question

1. Have families of distributions over orthonormal sets been defined and studied in the literature? What are a couple examples and/or references?
2. Are there known methods for constructing distributions over orthonormal sets by, for example, parameterizing the distributions of each coordinate of each vector?
3. As a follow-up to question (2) above, how does one simulate orthonormal sets by, for example, parameterizing some distribution over each coordinate and vector?

Consider an orthonormal set of dimension $n$. For question (3), I imagine one could define a procedure that begins by simulating a unit vector from a distribution of dimension $n-1$, and iteratively simulate from the remaining $n-2$ orthonormal basis vectors. However, beyond the first unit vector, how might one impose the mutually orthogonal constraint?

Answer 1

(Will update with details as I learn.)

1. Yes. A Haar measure is essentially a uniform distribution over the set of all orthogonal matrices. Orthogonal matrices of dimension $n$ are also known as orthogonal groups, denoted $\mathrm{O}(n)$.

2. and 3. Yes. For example, one could use the Gram-Schmidt process (GSP) which orthogonalizes a set of vectors in an iterative manner. This is a deterministic procedure, but randomness can be injected by initializing a set of random vectors and applying the GSP to these vectors. The resulting matrix will correspond to the Q matrix in QR decomposition. The Wikipedia article on orthogonalization lists other methods that can be leveraged in the same manner as above. For sampling from the Haar distribution, see How to generate random matrices from the classical compact groups (used in SciPy's ortho_group function).

Relevant references:

1. Technical report: The Subgroup Algorithm for Generating Uniform Random Variables
2. MathSE: "Random" generation of rotation matrices
3. MathSE: What does "Random Rotation Matrix" really mean? How to generate it?